M6R0 I+infsing/lnR6%%$fctG%$varG%#smG6&%'locvalG%$valG%(singminG%%exprG6"F-C&>8&9&>8 '-%#opG6#9$?(F-"""F9F-%%trueGC%>8$-%(infsingG6%F39%F0>8%7$-%-infsing/trieG6(-%1 infsing/infsolveG6&F3FA%&falseGF0&F=6#F9F7FA.%$logGFNFK@%0&FLFM%)infinityG@%/&& FCFMFMFS>F0-%$absG6#FR[FfnFCF-F-F-F- M6R0 I,infsing/expR6%%$fctG%$varG%(singminG6'%"rG%'devintG%"iG%%expsG%%cofsG6"F.C$>8 $-%(infsingG6%-%#opG6$"""9$9%9&@%0&F16#""#.%$logG-%'subsopG6$/F@.%*essentialGF1 C%>8%-%(dev/devG6%-%%subsG6$/F:*&&&F16#F8FVF8,&F8F8*$%&_XasyG!""FZF8F9F@""$@$/& FKFVF8C$>8(7#-%$seqG6$&FK6#,$8&F@/Fco;F8-%%iquoG6$-%%nopsG6#FKF@@%52F8-Fjo6#-%' selectG6%%%typeGF[o%%listG3-%'memberG6%%%trueG-%$mapGFcp.Fco0&FK6#,&FcoF8F8F8"" !-%'RETURNG6#FC>FK&FK6#Fco@%/FhnF@C$>8'7#-F^o6$&FK6#,&FcoF@F8F8Fdo@)0-%&minusG6 $<#-F66#F]r<$F8%)infinityG<"FC3-Fhp6%F8F]rF]q4-Fdp6$F`o%(integerG-FD6$/F@.%*alg ebraicGF1-Fdp6$F`o%'posintG-FD6$/F@.%&polarGF1-F"6%F9F:-%$absG6#FTFCF.F.F.F. M6R0 I,infsing/cosR6%%$fctG%$varG%(singminG6'%"rG%'devintG%"iG%%expsG%%cofsG6"F.C$>8 $-%(infsingG6%-%#opG6$"""9$9%9&@%0&F16#""#.%$logG-%'subsopG6$/F@.%*essentialGF1 C%>8%-%(dev/devG6%-%%subsG6$/F:*&&&F16#F8FVF8,&F8F8*$%&_XasyG!""FZF8F9F@""$@$/& FKFVF8C$>8(7#-%$seqG6$&FK6#,$8&F@/Fco;F8-%%iquoG6$-%%nopsG6#FKF@@%52F8-Fjo6#-%' selectG6%%%typeGF[o%%listG3-%'memberG6%%%trueG-%$mapGFcp.Fco0&FK6#,&FcoF8F8F8"" !-%'RETURNG6#FC>FK&FK6#Fco@%/FhnF@C$>8'7#-F^o6$&FK6#,&FcoF@F8F8Fdo@)0-%&minusG6 $<#-F66#F]r<$F8%)infinityG<"FC3-Fhp6%F8F]rF]q4-Fdp6$F`o%(integerG-FD6$/F@.%*alg ebraicGF1-Fdp6$F`o%'posintG-FD6$/F@.%&polarGF1-%,infsing/expG6%F9F:-%$absG6#FTF CF.F.F.F. M6R0 I,infsing/sinR6%%$fctG%$varG%(singminG6'%"rG%'devintG%"iG%%expsG%%cofsG6"F.C$>8 $-%(infsingG6%-%#opG6$"""9$9%9&@%0&F16#""#.%$logG-%'subsopG6$/F@.%*essentialGF1 C%>8%-%(dev/devG6%-%%subsG6$/F:*&&&F16#F8FVF8,&F8F8*$%&_XasyG!""FZF8F9F@""$@$/& FKFVF8C$>8(7#-%$seqG6$&FK6#,$8&F@/Fco;F8-%%iquoG6$-%%nopsG6#FKF@@%52F8-Fjo6#-%' selectG6%%%typeGF[o%%listG3-%'memberG6%%%trueG-%$mapGFcp.Fco0&FK6#,&FcoF8F8F8"" !-%'RETURNG6#FC>FK&FK6#Fco@%/FhnF@C$>8'7#-F^o6$&FK6#,&FcoF@F8F8Fdo@)0-%&minusG6 $<#-F66#F]r<$F8%)infinityG<"FC3-Fhp6%F8F]rF]q4-Fdp6$F`o%(integerG-FD6$/F@.%*alg ebraicGF1-Fdp6$F`o%'posintG-FD6$/F@.%&polarGF1-%,infsing/expG6%F9F:-%$absG6#FTF CF.F.F.F. M6R0 I-infsing/sinhR6%%$fctG%$varG%(singminG6'%"rG%'devintG%"iG%%expsG%%cofsG6"F.C$> 8$-%(infsingG6%-%#opG6$"""9$9%9&@%0&F16#""#.%$logG-%'subsopG6$/F@.%*essentialGF 1C%>8%-%(dev/devG6%-%%subsG6$/F:*&&&F16#F8FVF8,&F8F8*$%&_XasyG!""FZF8F9F@""$@$/ &FKFVF8C$>8(7#-%$seqG6$&FK6#,$8&F@/Fco;F8-%%iquoG6$-%%nopsG6#FKF@@%52F8-Fjo6#-% 'selectG6%%%typeGF[o%%listG3-%'memberG6%%%trueG-%$mapGFcp.Fco0&FK6#,&FcoF8F8F8" "!-%'RETURNG6#FC>FK&FK6#Fco@%/FhnF@C$>8'7#-F^o6$&FK6#,&FcoF@F8F8Fdo@)0-%&minusG 6$<#-F66#F]r<$F8%)infinityG<"FC3-Fhp6%F8F]rF]q4-Fdp6$F`o%(integerG-FD6$/F@.%*al gebraicGF1-Fdp6$F`o%'posintG-FD6$/F@.%&polarGF1-%,infsing/expG6%F9F:-%$absG6#FT FCF.F.F.F. M6R0 I-infsing/coshR6%%$fctG%$varG%(singminG6'%"rG%'devintG%"iG%%expsG%%cofsG6"F.C$> 8$-%(infsingG6%-%#opG6$"""9$9%9&@%0&F16#""#.%$logG-%'subsopG6$/F@.%*essentialGF 1C%>8%-%(dev/devG6%-%%subsG6$/F:*&&&F16#F8FVF8,&F8F8*$%&_XasyG!""FZF8F9F@""$@$/ &FKFVF8C$>8(7#-%$seqG6$&FK6#,$8&F@/Fco;F8-%%iquoG6$-%%nopsG6#FKF@@%52F8-Fjo6#-% 'selectG6%%%typeGF[o%%listG3-%'memberG6%%%trueG-%$mapGFcp.Fco0&FK6#,&FcoF8F8F8" "!-%'RETURNG6#FC>FK&FK6#Fco@%/FhnF@C$>8'7#-F^o6$&FK6#,&FcoF@F8F8Fdo@)0-%&minusG 6$<#-F66#F]r<$F8%)infinityG<"FC3-Fhp6%F8F]rF]q4-Fdp6$F`o%(integerG-FD6$/F@.%*al gebraicGF1-Fdp6$F`o%'posintG-FD6$/F@.%&polarGF1-%,infsing/expG6%F9F:-%$absG6#FT FCF.F.F.F. M6R0 I-infsing/trieR6(%#u1G%#u2G%$fctG%$varG%$ty1G%$ty2G6*%$resG%"uG%"vG%%res2G%"iG% "jG%$sigG%#tyG6"F4@+/9$7"-F"6(7#%)infinityG9%9&9'.%(polynomG9)/F=F8-F"6(F;F7F>F ?9(F@-%$hasG6$F7F<@%-%.infsing/checkG6&F=F>F?.8$6$FOFB6$F7FF-FH6$F=F<@%-FL6&F7F >F?FN6$FOFF6$F=FBC&>8%-%#opG6$"""F7>8&-Fgn6$FinF=>FO-%.comparemoduleG6$FenF[o@) /FO%"G@'FKFPFUFWFfo/FO%"=GC'FU-FL6&F=F>F?.8'?(8(FinFinF 41Fbp-%%nopsG6#FO?(8)FinFinF41Fhp-Fep6#F`pC$>8*-%)zerotestG6$&FO6#Fbp&F`p6#Fhp@ &/F^q%%trueGC$>F`p-%'subsopG6$/Fhp%%NULLGF`p>Fhp,&FhpFin!""Fin3/F^q%%FAILG-%%ty peG6$-%&evalfG6$-%&evalcG6#,&FdqFinFbqFbr"#5%(numericG@%2-%'lengthG6#Fbq-Fes6#F dqC$>F`pF[r>FhpFarC%>FO-F\r6$/FbpF_rFO>Fbp,&FbpFinFbrFin[@3/FFF@>8+FB/FBF@>FgtF F/FF.%&polarG>FgtFB/FBF[u>FgtFF/FF.%*algebraicG>FgtFB/FBFau>FgtFB/FF.%$logG>Fgt FB/FBFgu>FgtFF>FgtFF6$7$-FgnFfp-FgnF[qFgt-%&ERRORG6#%3Too~many~variablesGF4F4F4 F4, I.infsing/checkR6&%#ptGF'F(%'resultG6&F0F,%'devintGF1F4F4C%>FenF8?&FOF7Fhq@%-FH 6$FOF<>Fen7$-Fgn6#FenFOC$>F[o-%(dev/devG6%-%%subsG6$/F>*&FOFin,&FinFin*$%&_Xasy GFbrFbrFinF=""#""$@%/<#-%$seqG6$-Fgn6$FinF`p/F`p-%'indetsG6$F[o%%listG<#FinC$?( F`pFdxFcx,&-Fep6#F[oFin!"#Fin-Fgr6$&F[oF[q%*nonnegintGF4@$52F`pFfy33/F`pFfy4Fiy 2,&F[zFin&F[o6#,&F`pFinFhyFinFbrFin>FenFcw>FenFcw@%/FenF8%&falseGC$>F?FenFhqF4F 4F4F4 M6R0 I*infsing/QR6%%$fctG%$varG%(singminG6#%$valG6"F*C$>8$-%-infsing/trieG6(-%1infsi ng/infsolveG6&,&-%#opG6$"""9$F8!""F89%%%trueG9&7#%)infinityGF9F;.%&polarG.%(pol ynomG@%/&&F-6#F8FHF?-%(infsingG6%F5F;F=7$F-F8$9&?(F ,"""F2F,%%trueGC%>8%-%(infsingG6%-%#opG6$F29$9%F/>8&7$-%-infsing/trieG6(-%1infs ing/infsolveG6&-%$cotG6#F:F>%&falseGF/&F66#F2F=F>.%&polarG&F66#""#FK@%0&FLFM%)i nfinityG@%/&&F@FMFMFV>F/-%$absG6#FU[FinF@F,F,F,F, M6R0 I1infsing/LambertWR6%%$fctG%$varG%(SingminG6%%'locvalG%$valG%(singminG6"F,C%>8& 9&?(F,"""F2F,%%trueGC%>8$-%(infsingG6%-%#opG6$F29$9%F/>8%7$-%-infsing/trieG6(-% 1infsing/infsolveG6&,&F:F2-%$expG6#!""F2F>%&falseGF/&F66#F2F=F>.%*algebraicG&F6 6#""#FM@%0&FNFO%)infinityG@%/&&F@FOFOFX>F/-%$absG6#FW[F[oF@F,F,F,F, M6R0 I/infsing/RootOfR6%%$FctG%$varG%(SingminG6+%(singminG%'locvalG%$valG%"iG%"yG%$f ctG%$sigG%'tosubsG%"jG6"F2C&>8$9&>8)-%%subsG6$/%#_ZG8(-%#opG6$"""9$?(F2FBFBF2%% trueGC&@%4-%$hasG6$-%%diffG6$F89%F>>8%-%(infsingG6%FLFOF5>FQ7%7#%)infinityG.%(p olynomGFE>8&-%$mapG6%F:7#-%&solveG6$<$-FM6$F8F>F8<$F>FOFO?&8'-%'indetsG6$Ffn%'R ootOfGFE@$-%%typeG6$-F@6$FBFco-FZ6$%(numericGF=C$>8+7#-%$seqG6$/Fco-Fgo6$F\p8,/ F[q7#-%'fsolveG6%F\pF=%(complexG?&F[qFfnFE@$-FJ6$F[qFco>Ffn-F:6$/F[q-F@6#-Fhn6% F:FcpF[qFfn@%0-%%nopsG6#Ffn""!C%?(Fco""#FBF21FcoF`rC$>8*-%.comparemoduleG6$&Ffn 6#FB&Ffn6#Fco@&/Fjr%"Ffn-%'subsopG6$/Fco%%NULLGFfn>Fco,&FcoFB!""FB/Fjr%">G C$>Ffn7#-F@6$;FcoF`rFfn>FcoFB>Ffn7$-%-infsing/trieG6(Ffn&FQF_sFCFO.%*algebraicG &FQ6#Ffr%&falseG@%0&F]uF_sFX@%/&F^sF_sFX>F5-%$absG6#Feu[F]vF]vFfnF2F2F2F2 M6R0 I1infsing/infsolveR6&%%fctnG%"xG%(coefposG%(minsingG60%$fctG%$solG%$valG%$nbrG% "iG%"jG%%lpolG%%lracG%"kG%"lG%"fG%$auxG%%aux2G%$resG6"F8C-@%-%$hasG6$9$<$.%"LG. %"QG>8$-%%evalG6#-%%subsG6%/F@R6#F&F8F8F8-%#lnG6#*$,&"""FTF>!""FUF8F8F8/FBRFNF8 F8F8FRF8F8F8F>>FEF>@$353-%(hastypeG6$FE)%)anythingG%)fractionG4-Fhn6$FE%&floatG -F<6$-%$mapG6$%#opG-%'indetsG6$FE%'RootOfG9%-%%typeG6$-%(convertGFio-%'algfunG6 $F[oF[p-%'RETURNG6#-%8infsing/infsolve/algfunG6&FEF[p9&9'>8--%%igcdG6#-%$seqG6$ -Ffo6$""#8(/Fgq-Fho6$FE)-%*identicalG6#F[p%(integerG@$2FTF]qC$>8.-FJ6$/F[p)F[p* $F]qFUFE@$4-Fhn6$Fdr)F\rF\oC&>8+-F"6&FdrF[pFjp)F[qF]q>8'""!?&FgqFas%%trueG@%/Fg q%)infinityGC$>Ffs,&FfsFTFTFT>&8&6#FfsFgqC$?(8)FgsFT,&F]qFTFUFTFis>&Fbt6#,(FfsF TFTFTFftFT*&)FgqFirFT-%$expG6#**^#FfqFTFftFT%#PiGFTF]qFUFT>Ffs,&FfsFTF]qFT-Fep6 #7#-Fbq6$&Fbt6#Fgq/Fgq;FTFfs@$-F]p6$FE-%(polynomGFcp-Fep6#-%9infsing/infsolve/p olynomG6%FEF[pF[q>%1_EnvAllSolutionsGFis>8%-FG6#7#-%&solveG6$FEF[p@&-F]p6$F\w7# -%)specfuncG6$FcvFjo-Fep6#-Fgv6&-FJ6$/%#_ZGF[p-Ffo6$FT&F\w6#FTF[pF[qF>-F<6$F\wF joC$>8*7#-Fbq6$-Ffo6$FTFgq/Fgq-FhoFgx?(FgqFTFT-%%nopsG6#FjxFis@&-F]p6$&FjxF\vFc vC$>Fas-Fgv6$-FJ6$F`xFiyF[p?(FftFTFTF81Fft-Fdy6#F\w@$-F<6$&F\w6#Fft-Fjo6#FiyC$> F\w7$-Ffo6#-%'subsopG6$/Fft%%NULLGF\w-Fbq6$-FJ6$/Fiz8,Fgz/Fj[lFas>Fft,&FftFTFUF T/-%'selectG6$RFNF8F8F8-%&evalbG6#/-%#isG6$F>F_rFisF8F8F8-Fho6$Fgq%%nameG<"Z%C$ >8/-%'fsolveGF]z@$-F]p6$Fa]lF`oC$>Fas-%.numericRootOfG6$FiyFa]l?(FftFTFTF8Faz@$ Fez>F\w7$F^[l-FJ6$/FizFasFgzF8F8>F\w7#-Ffo6#-%&minusG6$<#-FfoFcz<#Fgs>Fa]l-%%ma p2G6%FfoFT-F`\l6$RFNF8F8F8-Fd\l6#/-Fdy6#F>FfqF8F8F8-Fho6$F\w-Fhw6$F[o%)LambertW G@'0F\w7"C'>Fa]l-%&unionG6$Fa]l-F`\l6$RFNF8F8F8Fc\lF8F8F8-Fho6$F\wF\]l@$0Fa]lF] ]l-Fep6#-%>infsing/infsolve/infinitelistG6%F\wF[qFa]l@$0F[qFgs>F\w-%=infsing/in fsolve/largerrootsG6$F\wF[q>F\w-%Fa]l-Fc]l6%FEF[p;FgsFT@%5/Fa] lFd[l4-F]p6$Fa]l%(numericGC&>Fas&-%(infsingG6%FEF[pFgsFex>Fa]lF``l?(FgqFTFT-Fdy 6#FasFisZ%@$/-%)zerotestG6#-FG6#-FJ6$/F[p&FasF\vFEFis>Fa]l7$-Ffo6#Fa]lFfdlF8F8Z %C$>80-Fc]l6%FEF[p;Fgs-%&evalfG6#-%$absG6#&FasFex@'5/F^elFd[l4-F]p6$F^elF`cl@%/ Fa]lF``l-Fep6#7#F\t-FepFjdl2-%&FloatG6$FT,&%'DigitsGFUFfqFT-Ffel6#F^el>Fa]lF^el FaflF8F8@$2-FfelFjdlFfflFafl>Fa]l-Ffal6$7#Fa]lF[q@%0Fa]lF``lC%>Fbt7#-Fjo6%FEF[p &Fa]lFex@$FjpC$>Fj[l-F`\l6%F<-Fho6#FEF[p@$-F]p6$Fj[l-%$setG6#F[rC$>81-F_q6#-Fbq 6$-Ffo6$FfqF]q/F]qFj[l@$2FTF\il>Fbt7#-Fbq6$*&&FbtFexFT-F_u6#**FbuFTFj[lFTFcuFTF \ilFUFT/Fj[l;Fgs,&F\ilFTFUFT-Fep6#FbtFaflFaflF86#FjvF8F8, I8infsing/infsolve/algfunR6&F*F&F'F(6'F4%$indGF.%"uG%,carefulsubsGF8F8C&>FE-%&n umerG6#-%'normalG6#-F`p6$F>Fjo>F\w7#-Ffo6#Fgo?(FbtFTFTF82FbtFbz@&4-F<6$-Ffo6$FT -Ffo6$FbtF\wF[pC$>F\w-Fa[l6$/FbtFd[lF\w>Fbt,&FbtFTFUFT-F<6$7#-Ffo6$;,&FbtFTFTFT FbzF\wFb\mC$>F\w7%-Ffo6$;FTFj\mF\wF^]mFb\m>FbtFj\m@%/F\wF``l-F"FipC'>FgqR6$F&Fj jlF8F8F8@'4-F-F]pFe[mF>-Fdo6%9!F>F[pF8F8F8?&FbtF\wFis>FE-%*resul tantG6%-FJ6$/FbtFfsFE-Fgq6$-Ffo6$FTFbtFfsFfs>FE-%$quoG6%FE-%$gcdG6$FE-%%diffGFb wF[p@$/-FJ6$/F[pFgsFEFgs>FE-%'expandG6#*&FEFTF[pFU-Fgv6&FEF[pF[qF>F8F8F8F8, I9infsing/infsolve/polynomR6&%$polGF&F(F*6%%"pGF+F.F8F8C)>FE-%(collectG6$F>F[p@ %1-%'degreeGFbwFfq>F\w-FawFdam>F\w-Fc]l6%F>F[p%(complexG>F\w-Fh^l6$<#F\wF\_l@$3 2Ffq9#0FjpFgs>F\w-Ffal6$F\w-Fcel6#Fjp@$2""$Ffbm>F\w-Fjal6%F\wF[qF[p@$Febm>F\wF^ bl@%5/F\wFcflFfamF\w7#-Ffo6#<#-Fbq6$-Fj]l6$-FJ6$/F[pFaxF>Fbt/FbtF\wF8F8F8F8, IFfsFgs>Fgq-.Ff elF^r?(FEFTFTFg_lFisZ%C$>F\w-Fcel6#-FJ6$/Fjp&F>FchlFgq@$3-F]p6$F\wF`cl2F\w$FT!" $C$>FfsF_t>FatFhemF8F87#-Fbq6$&FbtFchl/FEF^vF8F8F8F8, I=infsing/infsolve/largerrootsR6$F3F(6%F.F/F7F8F8C%>F\wFgs?(FEFTFTFg_lFis@$/-%. comparemoduleG6$FhemF[p%">GC$>F\w,&F\wFTFTFT>&FbtFczFhem7#-Fbq6$Fffm/FE;FTF\wF8 F8F8F8, I:infsing/infsolve/smallestR6#F+6&%#nbGF7F.F5F8F8@%/Fg_lFgsFcflC&>FEFT>Fdx&F>Fe x?(FbtFfqFTFg_lFisC$>Ffs-Fbgm6$Fdx&F>Fcjl@&/Ffs%"=GC$>FE,&FEFTFTFT>&F\wFchlF_im /FfsFdgmC$>FEFT>FgimF_im7#-Fbq6$&F\wFcjl/Fbt;FTFEF8F8F8F8, I>infsing/infsolve/infinitelistR6%%%gsolG%%miniG%%indsG64%$varG%"aG%"bGF5F&%"cG %"dG%#iaG%#ibG%%lposG%#raG%#rbG%%exprGF7%%gvarGF+%%expsG%'tosubsGF8F8C'>82-F`\l 6$RFNF8F8F8Fc\lF8F8F8Fjp>84-F`\l6%F<-Fdo6$Ffo-F`\l6%F]p-FhoFh_l-Fhw6$-%'linearG 6#FcuF_uFi[n@%0F^\nF]]lC%?&F^elF^\nFis@%-Fh\l6$*&^#FUFT-%&coeffG6%F^elFcuFTFT%% evenG>&85F\gl/-.F_uF\gl-F_u6#-FJ6$/FcuFgsF^el>Fi]nFd[l>83-FJ6$7#-Fbq6$Fi]n/F^el F^\nF>>Fi[n-Fd`l6$-Fh^l6$FjpFi[n-F`\l6$RFNF8F8F8Fc\lF8F8F8-Fho6$Fe^nF\]lC$>Fe^n F>>Fi[nFjp?&F^elFe^nFisC$>FE-%*intersectG6$Fi[n-Fho6$F^elF\]l@'/-FdyFchlFgs>&F\ ilF\glF^el52FTFc`n4-F]p6$F^el-Fi\n6#-FfoFchl>Fe`n-%Hinfsing/infsolve/infiniteli st/heuristicG6%F^elFEF[pC%>FEF]an>F\w-%&evalcG6#-Fe]n6%F^elFEFT@%/F\wFgs>Fe`nFb tC)>Fbt-Ffan6#-Fe]n6%F^elFEFgs>Fa]l-Fe]n6%Fbt^#FTFgs>Fj[l-Fe]n6%FbtFfbnFT>Fdr-F e]n6%F\wFfbnFgs>Fas-Fe]n6%F\wFfbnFT>Ffs-%&evalrG6#-%'SignumG6#,&*&-Ffel6#,&*&Fj [lFTFdrFTFT*&Fa]lFTFasFTFUFT-%%sqrtG6#,&*$FdrFfqFT*$FasFfqFTFUFTF[pFU@'/FfsFTC% >Fgq-Fbcn6#-%.evalr/intpartG6#,$*&,&*&FdrFTFa]lFTFT*&FasFTFj[lFTFTFTFadnFUFU@$4 -F]p6$FgqF_r>Fgq-FcelF\v>Fe`n6$,&*&F\wFTFgqFTFTFbtFT,&*&F\wFT,&FgqFTFTFTFTFTFbt FT5/FfsFU/FfsFgsC(>Ffs-F_dn6#,,**FasFTFj[lFTFdrFTFa]lFTFfq*&FdrFfqFj[lFfqFU*&Fd rFfqF[pFfqFT*&FasFfqFa]lFfqFU*&FasFfqF[pFfqFT>Fft,&F`enFUFaenFU>FjxFadn>Ffs7$*& ,&FftFTFfsFUFTFjxFU*&,&FftFTFfsFTFTFjxFU>F]q-Fdo6%%&shakeGFfsFjfl@%4-F]p6$F]q7$ -.%)INTERVALG6#%&rangeGF^hn>Fe`nFd[lC'>Ffs-%&floorG6#-Ffo6$FT-Ffo6#&F]qFex@%31F ihnFfs1Ffs-Ffo6$FfqF[in>F]q-Fa[l6$/FT,&FfsFTFUFTF]q>F]q-Fa[l6$/FTFfsF]q>Ffs-%%c eilG6#-Ffo6$FT-Ffo6#&F]q6#Ffq@%31FajnFfs1Ffs-Ffo6$FfqFcjn>F]q-Fa[l6$/FfqF_tF]q> F]q-Fa[l6$/FfqFfsF]q>Fe`n6$,&*&F\wFTF]inFTFTFbtFT,&*&F\wFTFejnFTFTFbtFT>Fe`nFd[ l-F_bl6#7#-Fbq6$Fe`n/F^elFe^nF8F8F8F8, IHinfsing/infsolve/infinitelist/heuristicR6%Fc[n%%varsGFfjm6(F.FijmF,F/F7%(nbte stsGF8F8C$>Fft""&@%/-FdyF^rFTC*>F\w-FfoF^r>Fbt7#-Fbq6$-Ffel6#-Fcel6#-FJ6$/F\wFE F>/FE;,$FftFUFft@$3-%'memberG6%-%$minG6#-FfoFcjlFbt.FE5/FE-FdyFcjl/FEFT-%&ERROR G6#%Acannot~find~smallest~singularityG>FE,(FEFTFftFUFUFT?(FfsFgsFT"#50-Fbgm6$-F J6$/F\w,&FEFTFfsFTF>FjpFdgmF8@$/Ffs"#6F]_o>FgqFh_o@%0FfsFgsC%?(FfsFTFTFd_o0-Fbg m6$-FJ6$/F\w,&FEFTFfsFUF>FjpFdgmF8@$F]`oF]_o6$FgqFg`oFgq-Fbq6$-Fh^m6%-FJ6$/-Ffo 6$FTF[pFEF>-Fa[l6$/FTFd[lF[pFjpF\^oF8F8F8F8 M6R0 I*infsing/LR6%%$fctG%$varG%(singminG6#%$valG6"F*C$>8$-%-infsing/trieG6(-%1infsi ng/infsolveG6&,&-%#opG6$"""9$F8!""F89%%%trueG9&7#%)infinityGF9F;.%$logG.%(polyn omG@%/&&F-6#F8FHF?-%(infsingG6%F5F;F=7$F-F8-7" >8,F??(8&""$""#-%%nopsG6#9$%%trueG@&4-%%typeG6$-%#opG6$FCFI%*nonnegintG>FA7$-FQ 6#FAFC-FN6$-FQ6$,&FC"""!""FgnFI%%listG>F>7$-FQ6#F>FC>8%""!@$0FAF?C(>8'-%$maxG6$ F`o-%$seqG6$,$&FC6#FDFhn/FC-%'selectG6%FN7#-Fjo6$FZ/FCFAFin@%-FN6$*&9%Fgn,&FeoF gnFgnFgnFhn%(integerG>8+,&FjpFgnFhnFgn>F_q-%&truncG6#-%&evalfG6#Fjp>8$-%>equiva lent/alglogeq/purjungenG6%FeoF_qF`o>8(-%&arrayG6%;F`oFeo;F`oF_q%'sparseG?(FCF`o FgnFeoFJC$>8**()FhnFCFgn-FQ6$,&FCFEFgnFgnFiqFgn-%*factorialG6#FCFgn?(8)F`oFgnF_ qFJC$>84-%&coeffG6%Fhr%#_XGFbs>&F^r6$FCFbs7$FE-Fjo6$-FQ6#7$-Fgs6%Fes%#_YG,&FCFg n85Fhn,&FgtFgnFCFhn/Fgt;F`oFC?&FCFAFJC$@%4FX@%54-FN6$FPF]q2F_q,$FPFhn>807&Fgn-F jo6$-FQ6#7$*&-%%subsG6$/%$_ALGFP-FQ6$FE&F^r6$F`oFbsFgnFZFgn,(FbsFgnFPFgnFgnFgn/ FbsFcrFgn,(F_qFgnFPFgnFEFgn>Fgu7&FgnFiuF`o%)infinityGC%>Fgu7#Fgn?(FbsF`oFgnF_qF JC%>81-%/dev/multbyrealG6$7%-FQ6#-F`v6$Fbv&F^r6$,$-FQ6$FDFZFhnFbsF`oF]w-FQ6$FEF Z?(82FEFgn-%%iquoG6$-FG6#FZFEFJ>Fdw-%(dev/addG6$Fdw-Ffw6$7%-FQ6#-F`v6$Fbv&F^r6$ ,$-FQ6$,&FexFEFgnFgnFZFhnFbsF`oF]w-FQ6$,$FexFEFZ>Fgu7%-FQ6#Fgu7#-FQ6$;Fgn,&-FG6 #FdwFgn!"#FgnFdwFhv@$0FguF`w>Fgu7%Faz7%FEFgn,&F_qFhnFhnFgnFjv@$F\[l>F_o-F]y6$F_ o-%*dev/cleanGFbz@$0F>F?C-@$33/FFFD/&FIF^pF`o/&&FI6#FEF^pFhn-%'RETURNG6#7'Fgn&F b\lFc\lFgnF`oF]w>86F`o?(83FgnFgn-FGF]o2Fj\l,&F[qFgnFEFgn@$-FN6$-FQ6$,&-FQ6$F\]l F>FgnFhnFgnFIFin>Fj\l,&Fj\lFgn-FQ6$FDFc]lFhn@$51F_]lFj\l2F]]lF\]l>F\]l,&F\]lFgn FhnFgn>F>&F>6#;FgnF\]l>Feo-Fgo6$F`o-Fjo6$F\p/FC-Fap6%FN7#-Fjo6$FZ/FCF>Fin>F\]l, (FEFgnF[qFgn-FQ6$Ff]lFIFgn>Fiq-F`v6$/FcvFhn-F[r6%FeoF\]l8.>F^r-F`r6%Fbr;F`oF\]l Fdr?(FCF`oFgnFeoFJC$>FhrFir?(FbsF`oFgnF\]lFJC$>FesFfs>F[tF]t?&FCF>FJC%>FguF`w?( FbsF`oFgnF\]lFJC%>Fdw-Ffw6$F]xFbx?(FexFEFgnFfxFJ>Fdw-F]y6$Fdw-Ffw6$FfyF\z>Fgu7% FazFdwFbs@$F\[lC&>8/FP>Fdw-Ff[l6#-F`v6$/F\`lF`oFgu?(FbsFgnFgn,&F[blFgnFgnFgnFJ> Fdw-F]y6$Fdw-Ffw6$-Ff[l6#-F`v6$/F\`lFbsFgu*&)FhnFbsFgn-%)binomialG6$FcblFbsFgn> F_o-F]y6$F_oFdwF_oF;F;F;F;, I>equivalent/alglogeq/purjungenR6%F&F/F26*%"fGF*F.%$polGF,F-F)%"tGF;F;C*>Fiq-%' seriesG6%-.%$expG6#,&*&,(9&FgnFhnFgn*$FisFhnFhnFgn-.%#lnG6#,&FgnFgn*&F_qFgnFisF gnFgnFgnFgnF_qFgnFisF_]l>F^r-F`r6%;F`o,$F[qFE;F`oF[qFdr?(FbsF`oFgnF[qFJC$>FC-FQ 6$,&FbsFEFgnFgnFiq?(F_oFbsFgn,$FbsFEFJ>&F^r6$F_oFbs-Fgs6%FCF_qF_o>FeoFgn>FhrFgn ?(F_oFgnFgn-%$minG6$F[qFIFJC$>Feo-%'expandG6#*&FeoFgn,(FcvFgn%#_UGFgnF_oFgnFgn> Fhr,&FhrFgn*&-%(convertG6$7#-%"$G6$.*&F`flFgn)FisFbsFgn/.Fbs;-Fgx6$,&F_oFgnFgnF gnFEF_o%"+GFgnFeoFgnFgn@%2F[qFIC$?(F_o,&F[qFgnFgnFgnFgn-Fhfl6$FIFeelFJC$>FeoF\g l>Fhr,&FhrFgn*&-Ffgl6$7#-Fjgl6$F\hl/F`hl;FbhlF[qFehlFgnFeoFgnFgn@$2FIFeel?(F_o, &FIFgnFgnFgnFgnFeelFJC$>Feo-F`v6$/)FaglF\jlF`oF\gl>FhrF`ilC$?(F_oF\jlFgnF[qFJC$ >FeoF_jl>FhrFcgl?(F_oFjhlFgnFeelFJC$>FeoF_jl>FhrF`il-F`dl6%*(-Fddl6#,$*&FaglFgn FetFgnFhnFgn-%&GAMMAG6#,&FaglFhnFcvFhnFhnFhrFgnFaglF\jlF;F;F;F;, I*dev/cleanR6#F%6'F)%%res2GF*F-%#lzGF;F;C&>FiqF`w>F^rF??(FCFEFEFFFJC$@%-FN6$FPF inC%>F_o7#FE?(FeoFEFE-FG6#FPFJ@$0-FQ6$FeoFPF`o>F_o7$-FQ6#F_o-FQ6$;FeoF\qFP@&3/- FGFc]mFD/-FQ6$FDF_oF`o>F_o-FQ6$FEF_o/F_oFh\m>F_oF`o>F_oFP@+0F_oF`oC$>Fiq7%-FQ6# FiqF_o-FQ6$,&FCFgnFgnFgnFI>F^rF?0FC,&FFFgnFhnFgn>F^r7#F[_m/F^rF?>Fiq7%Fi^mFgn&F I6#FF0F[_mF]w>Fiq7%Fi^mFgn-FQ6#F^r>Fiq7%Fi^mFgnF]w-Fe\lFj^mF;F;F;F; M6R0 I>equivalent/alglogeq/transfertR6$%$devG%"pG6(%"fG%"xG%$resG%"jG%"kG%#nbG6"F.C' @%3-%%typeG6$-%#opG6$""$9$%(integerG1F5""!>8)""#>F>""">8$-%'seriesG6%*$-%&GAMMA G6#,$8%!""FM/FLF5,(9%FAF>FAFAFA>8&F<@%/-F66$F?FCFFR-%(dev/addG6$FR7$F?-%$seqG6$-F66#7$*,-F66 $FXFfnFA)FM8(FA-%)binomialG6$,$FinFMF^pFA-F66$,&F^pF?FAFAFCFA-%*factorialG6#F^p FA,&F^pFAFinFA/F^p;F<,(FPFAF]oFAFinFM?(FXF?F?FYFhn>FR-Fao6$FR7$F?-Feo6$-F66#7$* ,F[pFAF]pFAF_pFA-F66$,&F^pF?FMFAFCFAFfpFAFip/F^p;FA,*FPFAFAFAF]oFAFinFM7%FAFR,& F5FAFAFAF.F.F.F. M6R0 I1equivalent/equivR6%%$fctG%"pG%"xG6*%%singG%'locfctG%$devG%%ndevG%"kG%$hhsG%%i nfsG%#tyG6#%)rememberG6"C'>8%9$>8*-%(infsingG6%F69&""!>8$&F96#""">8+&F96#""#@%3 0F@7"0&F@FB%)infinityG-%3equivalent/equsingG6(F6F@9%FSF=FEC+Z#@$-%%typeG6$-%'no rmalG6#F6-%(polynomG6$%)anythingGF=-%'RETURNG6#F>-%)userinfoG6%FC.%+equivalentG %7No~singularities~foundG@$-%$hasG6$F6%'RootOfG-%&ERRORG6#%*giving~upG>8&-%(dev /devG6%-%%subsG6$/F=*$%&_XasyG!""F6""$Fip@$5-FX6$F^p%)constantG/-%%nopsG6#F^pFi pF[o>8'F`q@$543-FX6$F^p-%%listG6#%*algebraicG-FX6$<#-%$seqG6$-%#opG6$,&8(FHFCFC F^p/Fir;FC-%%iquoG6$FdqFH-%$setG6#%(integerG-Ffo6$F^p^#FC-Fjo6$%1Not~regular~at ~0GF7>8)-%F^p-%Hequivalent/sa ddlepoint/HarrisSchoenfeldG6&F6FSF=FO@%/%0_NBSADDLEPOINTSGFC7%%-_saddlepointGFO F^p7%&F[uFBFOF^p/Fjs.%"HGC$>F^p-%>equivalent/saddlepoint/HaymanG6%F6F=FO@%-Ffo6 $F^pF[u@%Fht7%F[uFO-Ffr6$FHF^p7%F]uFOF[vF[v-FX6$-%)frontendG6$%'expandG7#-Fcp6% /%"QG*$,&FCFCF=FhpFhp/%"LG-%#lnG6#FivF6Fgn-F\o6#7#F>7%F[uFO-%=equivalent/saddle point/selleGFeuF3F3F3F3 M6R0 I4equivalent/alglogeqR6$%$devG%"pG6,%"iG%'resultG%+listjungenG%.listtransfertG% %totrG%"jG%#l2G%"sG%"kG%#l3G6"F2C,@$4-%%typeG6$9$%%listG-%'RETURNG6#""!@$2""#-% $maxG6#-%$seqG6$-%#opG6$"""8$/FL-%'indetsGF8-%&ERRORG6#%=Maybe~in~our~next~vers ion...G@$2-FI6$FKF9FK-F<6#-%5equivalent/H_HS_singG6#F9>8&7#FK>8'Fin@%/FVFK?(FLF AFA-%%nopsGFfn%%trueG@%4-F76$-FI6$FLF9F:@%0FL,&F_oFK!""FK>Fhn7%-FI6#FhnFfo-FI6$ ,&FLFKFKFKF9>8%7$F>F`pC%@%Fio>8+FfoC$@%2""$-F`o6#Ffo>Fip7#-FI6$;FK,&F^qFK!"#FKF fo>Fip7">Fdp7$,$-FI6$F^qFfoF[pF`p@$3-F76$F`p.%*nonnegintG-%'memberG6%F>7#-FF6$- FI6$,&8)FAFKFKFip/F]s;FK-%%iquoG6$-F`o6#FipFA.F]s>Fip-%'subsopG6%/,$F]sFA%%NULL G/F\sF\tFip@$0FipFhqC$>8(-%$mapG6%F7Fgr.%*nonposintG@%4-Fer6$%&falseGFbt>Fhn7%F ^pFipF`pC)>8*7#FA>8-Fbu>8,F>?(F2FKFKF2-Fer6%F\uFbtFesC&@$2FKF]s>Fau7$-FI6#Fau-F I6$;,&FfuFAFAFK,(FfuFAF]sFAF[pFKFip>Fdu7$-FI6#Fdu-FI6$;,&FfuFAF]sFA,(FfuFAF]sFA FKFKFip>Ffu,&FfuFKF]sFK>Fbt7#-FI6$;,&F]sFKFKFK-F`o6#FbtFbt>Fau7$F_v-FI6$;Fdv,(F fuFAFgwFAFKFKFip@$0FauFbu>Fhn7%F^pFauF`p@$0FduFbu>F[o7%-FI6#F[oFduF`pC&@%2F]qF_ o>Fip7#-FI6$;FK,&F_oFKFfqFKF9>FipFhq@%0-FI6$F_oF9%)infinityG>Fdp7$F>Fey>Fdp7$F> Fgy@$Fdr>FipFgs@$F_tC$>FbtFct@%Fit>Fhn7%FKFipF>C)>FauFbu>FduFbu>FfuF>?(F2FKFKF2 FhuC&>FauF^v>FduFgv>FfuF`w>FbtFbw>FauFjw@$F`x>Fhn7%FKFauF>@$Fdx>F[o7%FKFduF>>Fd p-%3equivalent/OtransfG6#Fdp@$0FhnFin>Fdp-%(dev/addG6$Fdp-%;equivalent/alglogeq /jungenG6$Fhn9%@$0F[oFin>Fdp-F]\l6$Fdp-%>equivalent/alglogeq/transfertG6$F[oFb\ lFdpF2F2F2F2, I3equivalent/OtransfR6#%%exprGF2F2F2@%0F9F[z@%-F76$FV<$F>%'negintG@%0FVF>7%FK7% FAFK,&FVF[pFKFK,&-FI6$FAF9FKFKFK7%FKFKF[^l7%FK7%FAFK,$FVF[pF[^lF>F2F2F2F2, I5equivalent/H_HS_singR6#F%6#%$resGF2F2C$>FL-%'dev/lnG6$F9FK@'33/FHFK2-FI6$F]qF LF>/-%&evalrG6#-%'SignumG6#-FI6$FAFLFK.%"HG1FHF>.%#HSG.%)pointcolGF2F2F2F2 M6R0 I3equivalent/equsingR6(%$fctG%%SingG%%palgG%%pdevG%"xG%)typesingG62%"iG%$resG%' devintG%'resintG%"fG%$ordG%)nextsingG%%singG%"pG%%multG%"jG%&mmultG%'remdevG%&f oundG%%rootG%#coG6"F8%""!>%0_EnvXasyintegerG%%trueG>8+9%@$-%%typeG6$9$-%(ra tpolyG6$%)rationalG9(C(>8(-%'normalG6#FK@$-FI6$FS%(polynomG-%'RETURNG6#F@>8,&-% (sqrfreeG6$-%&denomG6#FSFP6#""#>8/<#-%$seqG6$&8.Fao/FjoFin@%2"""-%%nopsG6#FdoC' >Fdo-%$maxG6#-%#opGFap?(FjoF^pF^pFdoFC>&8-6#Fjo%%NULLG?&8$FFFC?&FjoFinFC@$53-FI 6$F`q-%)specfuncG6$%)anythingG%'RootOfG0-%$gcdG6$&Fjo6#F^p-%%subsG6$/%#_ZGFP-Fh p6$F^pF`qF^p/-%'expandG6#-Fcr6$/FPF`qF`rF@C$>&F\q6#Fio6$FbsF`q[?(F`qFdo!""F^p/& F\q6#F`qF^qF<>FE7#-Fgo6$F[q/Fjo;-Fep6$F^p,(F`qF^p9&FgsF^pF^pF`q>F`qFgp@$33/FdtF ^p-%'memberG6$F^pFE19'FboC,>F\qF`q>8'7'F^p*$-%*factorialG6#,&F\qF^pFgsF^pFgs,&F \qFgsF^pF^pF^p,&F\qFgsFboF^p>Fin-%(factorsG6#-%(convertG6$-%$mapG6$R6#F)FFin7#-Fgo6$F`r/Fjo&FinFao@%2F^p-F`p6#FinC$?& F`qFEFC?&FjoFinFC@$53Feq0-F^r6$FjoFbrF^p/-F[s6#-Fcr6$F_sFjoF@C%>&81F]qF^p@%4-%) assignedG6#&82F]q>F_y<#F`q>F_y-%&unionG6$F_yFbyFes>Fin-Fcv6%R6$F)F9F&F`y6#-FhpFgwFE-%4equivalent/alglogeqG6%-%(dev/devG6%-Fcr 6$/FP,&F^pF^p*$%&_XasyGFgsFgsFSFboFboF^pF^p@$4-F[u6%F@-Fcr6$/FPF^pFin.FjoFen>Fi n7$&FinF]q-Fhp6#-%'subsopG6$/FjoF^qFin?&FjoFinFCC%>83*&)FgsF\qF^p-FU6#*(FSF^p)F joF\qF^p)-%%diffG6$FjoFPF\qFgsF^p>Fd\l*&-%$remG6%-%&numerG6#Fd\lFjoFPF^p-Fb]l6% -F_oFf]lFjoFPFgs?&F`qF_yFC>F?-%(dev/addG6$F?-%(dev/prdG6$-F`^l6$-Fcr6$F_sFd\lFb u-%(dev/expG6$-F`^l6$7'F^pF^pFgsF@%)infinityG-%'dev/lnG6$-%(dev/powG6$F`qFgsFbo F^p-Ffn6#F?>80-Fhp6$""%-Fhp6#Fhz?(F`qF^pF^p-F`p6#FEFCC(>FS-Fcr6$/FP*&&FEFjsF^pF ][lF^pFK@%33/9).%&polarG0Ff_lF^q-F]y6#&Ff_l6%FSFbo""$>8)-%$minG6$F^u,$-Fhp6$Fba l-FhzFaalFgs>Fdal-Fep6$F^uFbo>8&-Fhz6%FSFdalFdal>Fbu-Fez6$F`blFdt@(/Fbu.%"HGC$> Fbu-%>equivalent/saddlepoint/HaymanG6%-%%evalG6#-Fcr6%/%"LGRFfvFFbu-%Hequivalent/saddlepoint/HarrisSchoenfeldG6&-F`cl6 #-Fcr6%/FeclRFfvFFbu-%=equivalent/saddlepoint/selleG6%-F`cl6#-F cr6%/FeclRFfvFF?-F]^l6$F?-F`^l6$Fbu-%.dev/conjugateG6#-Fg^l6$-F`^l6$F[_l-F^ _l6$-Fa_l6$FihlFgsF^uFdt>F?-F]^l6$F?-F`^l6$Fbu-Fg^l6$-F`^l6$F[_l-F^_l6$-Fa_l6$F e`lFgsF^uFdt@%332F^u,&FdtF^pFboF^p-FI6$F?%%listG530&F?FarFC1-%+dev/lengthGFd_lF dt3/Ff[mFC1-Fep6#-Fgo6$-Fi[m6#-Fhp6$FboF`q/F`q-F^\l6$/F^pF^qF?Fdt@%0Fj`lF[al-F" 6(FKFEFdt,&F^uF^pF^pF^pFPFj`lZ%C$>8*-%(infsingG6%FKFP-%$absG6#Fihl@%330&Fa]mFar 7"0&F\^mFarF\_l0&Fa]mFao.%*essentialGC$@&3/Ff[mF^p0&F?6#FgsF\_l>F?-F^\l6%/!"#F@ /FgsF\_lF?3/Ff[mF@0&&F?6#F__mFj^mF\_l>F?-F^\l6%/F__m-F^\l6%/F__mF^q/FgsF^qFe_m/ Fgs6%Fi^mF@F\_lF?-F]^l6$F?-F"6'FKF\^m-%"$G6$,(FdtF^pFh[mFgsF^pF^pFboFPFa^mF?Fequivalent/saddlepoint/HaymanR6%%$fctG%$varG%%singG6)%"aG%"bG%"rG%"xG%'tosubs G%$auxG%"nG6"F0C(>8'9%>8$-%'expandG6#*(F3"""-%%diffG6$9$F3F;F?!"">8%*&F3F;-F=6$ F6F3F;@%3-%)assignedG6#%5_EnvSolveSaddlePointG/FK%&falseG>8&7"Z%>FO7#-%&solveG6 $/F68*F3F0>FOFP>8(-%>equivalent/saddlepoint/saddleG6(FO9&F?F4FMFX@%/Fen%%NULLG@ %0Fin%)infinityG7$%-_saddlepointG-%+dev/reduceG6$-%(dev/devG6%-%%subsG6$/F3*&Fi nF;,&F;F;*$%&_XasyGF@F@F;,$**F?F;""##F;Fbp%#PiG#F@FbpFBFepFcpF;F;Fbp7$Fao-Fco6$ -Ffo6%-Fio6$/F3F_pF`pF;F;FbpC$>Fen-%*dev/printG6%-%#opG6$F;FenF_pF_oZ'C$>8)-Fco 6$-Ffo6%*&-Fio6$/F3FenF`pF;-.%$expG6#,$*&F_pF;-%#lnG6#FenF;F@F;FbpFbpFbp@%0&Fiq 6#-%%nopsG6#FiqF_o7$""!Fiq7$Fcs-%(dev/prdG6$Fiq7'F;F;FcsF;F;Q;Not~a~convergent~ expansionF0C%-%&printG6#%8,9%@--%%typeG6$9$-%(polynomG6$%)any thingGF6%%FAILG-F:6$F<%"*GC&>8%""">8&FA?(8'FHFH-%%nopsG6#F<%%trueGC$>8--%#opG6$ FLF<@%-F:6$FSF=>FG*&FGFHFSFHC$>FS-F"6$FSF6@%0FSFA@$0FSFJ@&/FS.%#HSG>FJF`o3/FS.% "HG/FJFA>FJFeoFA@%30FJFA/-%&evalrG6#-%'SignumG6$-%'lcoeffG6$-%'expandG6#FGF6""! FHFJFA-F:6$F<%"^GC$>FS-FU6$""#F<@'-F:6$FS%'posintG-F"6$-FU6$FHFFA- F"6$-.%$expG6#*&FSFH-%#lnG6#FhqFHF6-F:6$F<.-F`r6#F@C$>8(Fhq@&-%'memberG6$-F"6$F ]sF6<$F`oFeoF`o-F:6$F]sF=C$>8)-%'degreeGFcs@%0-F^p6#-Fap6#-FdpFcsFHFAC%>FJFip?( FLFaqFHFisF[pC$>8$Fis?(F3FHFHF332FHFhtF[p@%0-%%modpG6$FhtFLFip>Fht,&FhtFH!""FHC $>FS-F^p6#-Fap6#-%&coeffG6%F]sF6Fht@)/FSFip>FhtFbu/FSFcu>FJFA/FSFH>FhtFH-%&ERRO RG6#%FJFA>8*Fip>8+7#Fip?(F LFHFHFMFPC$>FSFT@$4FXC$>8.Fhn@'/FjwF`o@%/FJF`o@$/-%,dev/compareG6$-%(dev/devG6% -%%subsG6$/F6%&_XasyGFSFHFHFawFH>FawFSC$>FJF`o>FawFdx/FjwFeo@&/FJFeo@$F`x>FawFS FgoC$>FJFeo>FawFdx>F_w,&F_wFHFSFH@$F[p@%/-Fbx6$Faw-Fex6%-Fhx6$FjxF_wFHFHFHFJFAF AF3F3F3F3 M6R0 I>equivalent/saddlepoint/saddleR6(%"lG%%singG%$fctG%$varG%(printitG%"nG6*%"iG%' tosubsG%"jG%"rG%$ptsG%-alreadytriedG%$resG%$sigG6"F4C&>8'9$@%332""!-%%nopsG6#F7 2-%'lengthGF@"$+"4-%$hasG6$F7%'RootOfGC&?(8$"""FMF41FLF>C$>8*-%(dev/devG6%-%%su bsG6$/9)%&_XasyG-%#opG6$FLF7FMFM@'354-%%typeG6$FQ%%listG/-Ffn6$""$FQF=09%%)infi nityG@%53F[o/-%)zerotestG6$FeoFQ%%trueG/-F\p6$-Ffn6$""#FQFeoF^p@$2FMFL@%/-F\p6$ -Ffn6$FMF7FenF^pC$>F7-%'subsopG6$/FL%%NULLGF7>FL,&FLFM!""FM?(8&FMFM""&F^pC$>8+- %,dev/compareG6$-FS6%FUFhqFhq-FS6%-FV6$FXF[qFhqFhq@&/F\rFfqC%>F7-F`q6$/FMFcqF7> FLFeq>FhqFiq/F\rFMC%>F7F_q>FLFeq>FhqFiqC$>F7F_q>FLFeq532FaoF=/-%&evalrG6#-%'Sig numG6#-%+dev/lcoeffG6#FbpFM333F`o-F]o6$FbpF_o2-Ffn6$FcoFbpF=/-F\t6#-F_t6#-Fbt6# -Ffn6$FdpFbpFM@$Ffp@%FhpC$>F7F_q>FLFeq?(FhqFMFMFiqF^pC$>F\rF]r@&F_sC%>F7Fjr>FLF eq>FhqFiqFgrC%>F7F_q>FLFeq>FhqFiqC$>F7F_q>FLFeq>8(-%$mapG6&FS-FV6$FXF7FcoFco>8% 7"?(FLFMFMF41FL-F?6#Fjv@%54-F]o6$&Fjv6#FLF_o/&F\x6#-F?6#F\xFfo>Faw7$-Ffn6#FawF\ x@'/-Ffn6$FMF\xFM@%53/FeoFfo2FM,&F_xFM&F\x6#FcoFfq3Fdo2FMF_x>FawFdxC$>Fjv-F`q6$ /FL-FS6%FUFiqFiqFjv@'F\y>FawFdx/FewFMC%-%&printG6$%5The~saddle~point~is~GF[q-Fb z6#%:Saddle~point's~expansion:G-Fbz6#-%*dev/printG6%&Fjv6#FMFYFiqC%-Fbz6#-%$cat G6&%%The~GFL%4th~saddle~point~is~GFen-Fbz6#%2Its~expansion~is:G-Fbz6#-F[[l6%F\x FYFiqF_zC%-Fbz6$Fdz&F7F^[lFez-Fbz6#-F[[l6%F][lFYFcoC%F`[lFg[l-Fbz6#-F[[l6%F\xFY FcoC&>Fjv-%@equivalent/saddlepoint/pointcolG6&9&9'FeoFco>FawFbw>8)%&falseG@%0Fj v%%FAILG?(FLFMFMF4Fdw@'/&F\xF^[lFM@)349(55F^xF]y5Fcy3/F_xFM-F]o6$&F\x6#,&FaxFMF fqFMF_o>FawFdx33F_^l2FcoFax4Fd]lC%>Fd]lF^p>FQ-F^]l6&F`]lFa]lFeoFiq@'53F^y2FM,&& &FQF^[l6#-F?6#F[`lFM&F[`lFbyFfq3Fdo2FMFj_lC%>FawFbw>FjvFQ>FLF=F_zC$FezFb\lC$-Fb z6#-Fc[l6%%5Saddle~point~number~GFL%":GFg\lF_zC$FezFb\lC$Fh`lFg\lF_zC$FezFb\lC$ Fh`lFg\l-Fbz6#%FWarning:~saddle~point~value~not~foundG>%0_NBSADDLEPOINTSGFew@%0 FawFbwFawFcqF46#FfalF4F4 M6R0 IHequivalent/saddlepoint/HarrisSchoenfeldR6&%$fctG%"pG%$varG%%singG64%"bG%"gG%" cG%"uG%"dG%"iG%"jG%"kG%"lG%$resG%&sommeG%"sG%"qG%"mG%%prodG%"xG%$auxG%"nG6"F839&>&8+6#""!FE>84-%'normalG6#*&-%%diffG6$9$F?"""FO!""?(8)FPFP,&9%""#FVFP%%tru eGC$>&8$6#FS*(FGFP)F?FSFP-%*factorialGFfnFQ>FG-FM6$FGF?>8&,$*&F?FP-FM6$&Fen6#FP F?FP#FPFV@%3-%)assignedG6#%5_EnvSolveSaddlePointG/F\p%&falseG>8'7">F`p7#-%&solv eG6$/Fdo,&85FPFPFPF?-%>equivalent/saddlepoint/saddleG6(F`p9'FOF@F^pFip@%0F]q%)i nfinityGC$>8(-%*dev/printG6%-%(dev/devG6%-%%subsG6$/F?*&F]qFP,&FPFP*$%&_XasyGFQ FQFPF_oFUFUFarF`q?(FSFPFP-%$maxG6$FVFUFW>&8%Ffn-Fhq6%-F[r6$F]r,$*&,&&Fen6#,&FSF PFVFPFP*()FQFSFPFbsFQFdoFPFPFPFcqFQFQ-Fdr6$FP,&FUFPFSFQFesC$>Fcq-Feq6%-Fhq6%-F[ r6$/F?FarF_oFUFUFarF`q?(FSFPFPFcrFW>Fgr-Fhq6%-F[r6$F`tF]sFesFes>8-FP?(FSFPFP-Fd r6$FP-%&truncG6#,$FUFfoFWC%>8.FE?(8*FPFP,$FSFVFWC(>&FCFeo,(FSFVFduFQFPFP>8/FE?( 8,FVFPFduFW>&FC6#F]vFP>81FP?(F82FP?(80FPFPFduFW>Fiv-%(de v/prdG6$Fiv&Fhr6#&FC6#F[w>F[v-%(dev/addG6$F[vFiv>Fbv,&FduFPFQFP?(FFbv,&FbvFPFQFP>F\x,&F\xFPFQFP>&FC6#,&FbvFPFPFP,&&FC6#FduFPFPFP>FgxFP>Fbu- Ffw6$Fbu-%/dev/multbyrealG6$F[v*&-%&GAMMAG6#,(FduFPFSFPFfoFPFP-FjnFhxFQ>Fht-Ffw 6$Fht-F^y6$-F^w6$Fbu-%(dev/powG6%Fcq,$FSFQFes*&FdsFP-%%sqrtG6#%#PiGFQ@%F_q-F^w6 $-Ffw6$Fht-%&dev/OG6#-F^z6%FcqF`zFP-Fhq6%-F[r6$F]r,$*(FOFPFbzFQFcq#FQFVFfoFUFU- F^w6$Fiz-Fhq6%-F[r6$F`tFd[lFUFUF8$ -%$maxG6$""$9'@%09&%)infinityG>8%-%-dev/implicitG6%-%(dev/devG6%-%%subsG6$/9%*& F7""",&FGFG*$%&_XasyG!""FKFG*(FEFG-%%diffG6$9$FEFGFPFKF/F/7'FGFGFK""!F8F/>F:-F< 6%-F?6%-FB6$/FEFJFLF/F/FQF/@%0F:%%FAILG@%/F7F8F:-%$mapG6%%/dev/multbyrealG-F[o6 %%(dev/addG-F[o6&%(dev/powGF:FKF4FK,$F7FKFgnF,F,F,F, M6R0 I=equivalent/saddlepoint/selleR6%%$fctG%$varG%%singG6&%"uG%"hG%"xG%"nG6"F-C)>8& 9%>8%,&-%#lnG6#9$"""*&,&8'F9F9F9F9-F66#F0F9!""@%3-%)assignedG6#%5_EnvSolveSaddl ePointG/FE%&falseG>8$7">FI7#-%&solveG6$/-%%diffG6$F3F0""!F0-%>equivalent/saddle point/saddleG6(FI9&F8F1%%trueGF<-%&printG6#%enAssuming~the~saddle~point~method~ is~valid~in~this~context,G-Fen6#%/the~result~is:G@%0FX%)infinityG6$%0_NBSADDLEP OINTSG-%+dev/reduceG6$-%(dev/devG6%-%%subsG6$/F0*&FXF9,&F9F9*$%&_XasyGF?F?F9,$* *F8F9""##F9F`p%#PiG#F?F`p-FR6%F3F0F0FcpFapF9F9F`p6$F_o-Fao6$-Fdo6%-Fgo6$/F0F]p, $*,F8F9)F0F8$-%'signumG6%""!,&9$"""9%!""F2@+/F.F5%">G/F.F7%"F+F+F+F+ M6R0 I+equivalentR6&'%$fctG%*algebraicG'%"xG%%nameG'%"nGF*'%%optpG%'posintG6$%"pG%$r esG6#%ECopyright~Bruno~Salvy,~INRIA,~FranceG6"C'@%/9#""%>8$9'>F<"""@$4-%#isG6$9 &%(integerG-%-additionallyGFD@$/-FC6#2"$+"%&_XasyG%%FAILG-FHFL>8%-%1equivalent/ equivG6%9$F<9%@%4-%$hasG6$FS%-_saddlepointG-%*dev/printG6%FSFEF<@)0-%#opG6$""#F S%)infinityG@'/-%%nopsG6#FS""$-%%evalG6#-%%subsG6%/FX*$,&F?F?*&-F_o6$F?FSF?F^o! ""FepFep/%"OG-F]p6%/%%_RESGFS/%#_nGFER6#F)F5F5F5-Fgp6#*&FWF?)-F_o6$F?F[qF]qFepF 5F5F5-Fjn6%-%/dev/multbyrealG6$-F_o6$FhoFS*$)FcpFEFepFXF8%9$@'-%%typeG6$F>.-%(complexG6#%(numericG@)/F>""!-%%subsG 6$/.%,_Envsignum0GFK-%%evalG6$FQF5-FB6$F>.FH-%%signG6#F>-FB6$F>.-FF6#%&floatG-% &evalfG6#-F\o6$*&F>F5-%$absGFZF8,&%'DigitsGF5""#F5F`o4-%)assignedG6#FQ-FM6$/FQF K-%,signum/mainG6$F>FQC$>FQ-%'freezeG6#FR-%%thawG6#F]p/F;FeoC%>8$F?>F>9%@)/F[qF K-F"FZ-FB6$F[q.%'posintG@)FJ-%&ERRORG6#%Bsignum~is~not~differentiable~at~0G55FU -%'memberG6$-F"6%FKF>FK<$F8F53/-%#ImGFZFK/-%*traperrorG6#-%#isG6#0F>FK%%trueGFK FA-Fgq6#%Osignum~is~not~differentiable~at~complex~valuesG-.F"6$F5F>-FB6$F[qFD-F gq6#%Fnumeric~1st~argument~should~be~0~or~1G-Fcs6$F[qF>/F;""$C%>F[qF?>F>F]q@)F_ q@%FA@'FJ9&FUFXF`oC%>FQ-Fcp6#Fdt>8&-F"6#F]q@%5/Fjt-FfpFio-FB6$Fjt.%)constantGFj t-Ffp6#-FS6#-FM6$/F"R6#F06"6$%)operatorG%&arrowGF^v-Fcs6%FKF?FQF^vF^vF^vFjtFaqC %>FQFgt>Fjt-F"Fds@%/FjtFbs-Fcs6%F5F>F`uFjtFesFgs-Fcs6#9"-Fgq6#-%$catG6$%Dexpect ing~1,~2~or~3~arguments,~got~GF;F^vF^vF^v%*protectedG, I,signum/mainR6$F0%$EnvG6-F1%"sG%"tG%"fG%"wG%#xrG%#xiG%#sgG%"kG%$sgpG%$sgnGF2F^ vC)@0-FB6$F?.%&rangeGC$>F>-%$mapG6$F"F?@%55/F>;F5F5/F>;F8F8/F>;FKFK>F[q-%#opGFd s>F[q-Fcs6#F?-FB6$F?%"*GC&>F>F5>FjtF5?&8(F?F^sC$>8+-F"6#Ffz@%-%$hasG6$FizFcs>F> *&F>F5FfzF5>Fjt*&FjtF5FizF5@'5/F]qFK/F]qFP@&/F>F5>F[qFjt0F>F?>F[q*&FjtF5F`qF5/F jtF]q@&Fi[l>F[qFjtF[\l>F[qF]\lC$>F>-F^y6$RF]v6#F[xF^vF^vC$>F[q-F"6%FKF?FK@%33-F B6$F[qFcu0F[qFK/-%'indetsG6$F[q.%)functionG<"F[qF?F^vF^vF^vF?@$F[\l>F[qF`q3-FB6 $F?%"^G-FB6$-F[z6$FeoF?FW@(3-FB6$Fa^l.%%evenG3/-Fer6#-F[z6$F5F?FK5-F]r6$F]q<$F5 FP/-Fhr6#-F[s6#0F]_lFKF^s>F[qF53-FB6$Fa^l.%$oddG3Fj^l5-F]r6$F]q<&F8FKF5FPFc_l>F [q-F"F\_l-F]r6$F]q<%FKF5FP>F[q)Fe`lFa^l33F\^l/Fe`lF5/-Fer6#Fa^lFK>F[qF53333-FB6 $F?.-%(polynomG6#%)rationalG/-%%nopsG6#-Fd]lF^zF5/-%'degreeGF^zF50,&-Fhr6#-%'lc oeffGF^zF5F8F5FK5-F]r6$F]q<$FKFP0,&-FboFhblF5F8F5FKC$>Fjt-F[zF`bl@%F\cl>F[q*&-F "FhblF5-F"6#,&FjtF5*&-%&coeffG6%F?FjtFKF5FiblF8F5F5>F[q-F"6#,&*&FhclF5FjtF5F5*& F]dlF5FaclF8F533-FB6$F?%"+G/-Fhr6#-FYF^zF8F\cl>F[q,$-F"6#,$F?F8F83-FB6$F?Ff]l-F B6$-F[z6$FKF?.%'symbolGC$>8'-Fbw6$%(signum/GFiel@$-FB6$F_fl.%*procedureGC$>F>-F _fl6#-F[zF^z@$0F>%%FAILG>F[qF>@$34-Fho6#F[qFhdlC$>Fjt5-F]r6%F67#F\gl.8,-F]r6%F7 F[hlF\hl@%3Fjt/-Fhr6#-F[s6$-%'subsopG6$/F]hlFKF?%%realGF^s>F[q-F"6#-F[z6$F]hlF? C%>F>Fh]l?&FjtF?1-F_blFZF5>F>-%&minusG6$-%&unionG6$F><#-F"6#Fjt<#FK@$/FeilF5>F[ q&F>6#F5@$33Fcgl-FB6$F?.%)realconsG4-F^[l6$F?^#F5C$>F>-%-evalr/SignumGF^z@(/F>- .%)INTERVALG6#Fhy>F[qF]q5/F>-Fh[m6#Fdy/F>-Fh[m6#Ffy>F[q-F[z6$F5-F[zFZ-FB6$F>.-F i[m6#;FHFH-%'RETURNG6#F]z@$FdglC$@%-FB6$F[qFgn@%/-%%fracGFeglFK-%&truncGFeglF[q >-9!F]wF[q-F^]mFegl@$3/-Fhr6#-F[s6$F?F[ilF^s-F]r6$-Fhr6#-F[s6#2FKF?<$%&falseGF^ sC$>8-Fj^m@%/Fa_mF^s>F[qF5C$>8.-Fhr6#-F[s6#2F?FK@,/Fg_mF^s>F[qF8533/Fa_mF^_m/Fg _mF^_m-FB6$F?Fcu/-Fhr6#-F[s6#/F?FKF^s@%Fg[l>F[qFK>F[qF]qFg[l@&/-Fhr6#-F[s6#1FKF ?F^s>F[qF5/-Fhr6#-F[s6#1F?FKF^s>F[qF83/F]qF5Faam>F[qF53/F]qF8Fham>F[qF8@$3FcglF ^[mC$>8)-%#ReGF^z@$34-F^[l6$Fibm<$FerF[cm0FibmF?C$>8*-FerF^z@$4-F^[l6$FecmFacm@ (45-FB6$FibmF6-FB6$FecmF6C$@'/FibmFK>F>-Fbo6#Fecm/FecmFK>F>-Fbo6#Fibm>F>-%%sqrt G6#,&*$FibmFeoF5*$FecmFeoF5@&FJ>F[qF]q5Fg[lFfr>F[q-%'normalG6#*&F?F5F>F84F`dm>F [q-F"6#-FM6$/F6F5Fibm4F^dm>F[q*&F`[mF5-F"6#-FM6$FafmFecmF5@'Fcgl>F\^mF]zFc]m@%F f]mFi]mF[q>F\^mF[qF^vF^vF^v6" M6R0 I(infsingR6%%$fctG%$varG%"rG6-%$valG%"iG%(minsingG%(singminG%)singminiG%'locval G%&exponG%(rootvalG%&type1G%&type2G%"fG6$%)rememberG%ECopyright~Bruno~Salvy,~IN RIA,~FranceG6"C$@%/9#""$>8&9&>F>""!@--%%typeG6$9$-%(polynomG6$%)anythingG9%7%7# %)infinityG.FH-FD6$-%'expandG6#FF-FH6$.%,nonnegativeGFK-FD6$FF-%(ratpolyGFI7%-% 1infsing/infsolveG6&-%&denomGFTFK%&falseGF>.%&polarGF]o-FD6$FF<$%"*G%"+GC%>8$7% FMFO%%trueG?&8%FFFioC$>8(F>?(F7"""F`p"#5FioC$>8)-F"6%F[pFKF^p@%0&&Fdp6#F`pF[qFN C$>Fgo7$-%-infsing/trieG6(Fjp&FgoF[qFFFK&Fdp6#""#&FgoFdq-%&evalbG6#3&Fdp6#F<&Fg oF\r@%5/&FbqF[qFN/-%.comparemoduleG6$FarFip%">G>F^p-%$absG6#Fip[F[sFgo-FD6$FF%" ^GC$>8*-%#opG6$FeqFF@%-FD6$Fas%)constantG@%-FD6$Fas%'posintG-F"6%-Fcs6$F`pFFFKF >C&>FgoFho>8'F>?(F7F`pF`pF7FioC'>8+-Fin6&F_tFKF]oFdt>Fdp-F"6%F_tFKFdt@%4-FD6$Fa s%'negintGC$@%/FcqF^o>8-.%*algebraicG>FguFcq>8,FhuC$>F\vF^o>FguFcq>Fgo7$-F`q6(F htFjpFFFKF\vFguF]o@%Fhp@%F`r>FdtFhrF[sF[sFgo-F"6%-%$expG6#*&FbsF`p-%#lnG6#F_tF` pFKF>-FD6$FF%)functionGC$>8.-%$catG6$%)infsing/G-Fcs6$FAFF@%4-FD6$Few%*procedur eG-%&ERRORG6$%=Not~implemented:~infsing~of~GFjw-Few6%FFFKF>-Fbx6$%3Invalid~expr essionGFFF7F7F7F7 M6R0 I&evalrR6#%%exprG6'%$varG%"rG%"iG%"jG%"uG6#%aoCopyright~(c)~1991~by~the~Univers ity~of~Waterloo.~All~rights~reserved.GE\s#-%$minG6$"""%)infinityGF2-%'SignumG6# ""!F7@%34-%(hastypeG6$9$%&rangeG/-%%nopsG6#-%*intersectG6$-%$mapG6$R6#%"xG6"FMF M-%#opG6$F7F>FMFMFM-%'indetsG6$F>%)functionG<).F0.%$maxG.%$absG.F5.%&shakeG.%$I ntG.%$intGF7F>C'>8%-%*traperrorG6#-%,evalr/evalrG6$F>%%trueG-%'subsopG6$/""%%%N ULLG-%%evalG6#Fco@$/F^o%*lasterrorG-%'RETURNG6#%%FAILG@$-%$hasG6$F^oF3>F^o-%2si mplify/infinityG6#F^o@%5-%%typeG6$F^o.%%listG-Faq6$F^o-%)specfuncG6$%)anythingG .%)INTERVALGC$?(8'F2F2-FBF]qFeoC$?(8(F2F2""#Feo@$4-Faq6$-FO6$7$F_rFcrF^o%)const antGC$>8$-%*evalr/varG6#Fir?(8&F2F2-FB6#F_sFeo@$/-FO6$7$FdsFdrF_s;,$F3!""F3>F^o -Fgo6$/F_r-Fgo6$/Fcr-%%subsG6$/-FO6$FdsF_s-FO6$7$FdsF2F_sFir-FO6$F_rF^oF^o@$/-F O6$7$F_rF2F^o-FO6$7$F_rFdrF^o>F^o-Fgo6$/F_rFcuF^o@%4-F<6$F^oF?-FOF]qF^oF^oFMFMF M%*protectedG M6R0 I,type/RootOfR6#%"xG6"6%%)rememberG%'systemG%aoCopyright~(c)~1990~by~the~Univer sity~of~Waterloo.~All~rights~reserved.GF&@%/9#"""33-%%typeG6$9$%)functionG/-%#o pG6$""!F4.%'RootOfG-F26$7#-F86#F4<&7#%*algebraicG7$FD%)constantG7$FD%&rangeG7$F D%)equationG%&falseGF&F&F&6" M6R0 I'RootOfR6$%"eG%"xG6.%#exG%%inamG%"zG%'freezeG%$rofG%%rofsG%"iG%#lcG%%indsG%$ex 1G%$errG%'indexdG6$%)rememberG%aoCopyright~(c)~1992~by~the~University~of~Waterl oo.~All~rights~reserved.G6"C7@*31""#9#-%%typeG6$9%%%nameG@)30FA%#_ZG0-%*interse ctG6$-%'indetsG6#9$<#FF<"-%&ERRORG6#%Jshould~not~use~_Z~as~independent~variable G-%$hasG6$-%'selectG6%F?FKF"FAC%>8,FX@$33/-%%nopsG6#Fgn"""-F?6$FN-%(polynomG6$% )anythingG&Fgn6#F_o-F?6$-%#opG6$F_oFfo-Fco6$FeoFFC$>8$-%*resultantG6%-%%subsG6$ /Ffo8&FN-Ffp6$/FFFipFjoFip-%'RETURNG6#-F"6$Fap&9"6#;FFhu%&falseG@&33-F?6$FN%"^G/-%'signumG6#FftF_o4Fh u-F^q6#-F"6$FdtFbq3-F?6$FN%"*G-F?6$Fdt%(numericG-F^q6#-F"6$-%'subsopG6$/F_oF_oF NFbq>Fap-%*traperrorG6#-%&numerG6#-%'normalGFM@$/Fap%*lasterrorG-F^q6#.-F"6#Fcq >8%-FY6%F?-FL6#FapFB@$3/-F]o6#F[yF_o0-F[p6$F_oF[yFF-F^q6#-F"6%FapFfyFbq@$4-FV6$ FapFFF^u@$0FapFNF]q@$3/%2_EnvRootOfComposeG%%trueGFev?&8(-FL6$Fap-%)specfuncG6$ FeoF"Ffz@$3-F?6$-F[p6$F_oFhzF]p-F?6$Fap-Fco6$FeoFhz-F^q6#-F"6$-Fcp6%-Ffp6$F\qFb [l-Ffp6$/FhzFipFapFipFbq>8)Fiz>8'7#-%$seqG6$/&Fd\l6#8*&FhzF]]l/F^]l;F_o-F]o6#Fd \l>Fap-Ffp6$Ff\lFap@$4-FV6$Fe]lFFF^u@$-FV6$-FL6#<#-%)frontendG6$%'coeffsG7$-%(c ollectGF_zFFFFFex>Fap-Fb^l6%Ffp7$F\qFN7$<$F]w%"+G<#F\q@$/-%'degreeG6$FapFipF_oC $@$Fhu-%0RootOf/chkindexG6$-F[p6$F8--F[x6 #-%&evalaG6#-%'NormalG6#,$*&-%&coeffG6%FapFip""!F_o-F`al6%FapFipF_oFhtFht@%0Fd` lFdxC$@$3/F=F<-F?6$Feu;F`wF`w@%-%#isG6$Fd`l-%*RealRangeG6$-F[p6$F_oFeuFj_l-F^q6 #Fd`l-F^q6#%%NULLGFgbl-FR6#%Ileading~coefficient~should~be~invertibleG-F?6$Fap- Fco6$FeoFipC%>Fd`l-FaxF\al@$Fial@%F_blFgblFiblFgbl@$-F?6$Fap-Fco6$F]uFipC*@$-FV 6$Fap.F"C$>Fd`l-F[x6#-Fh`l6#-%)PrimpartGFd_l@&FfalC$>FapFd`l@&2Fb_lF<-F^q6#-F"6 $-Ffp6$/Fip.FFFapFbq4-F?6$-%'lcoeffGFd_l.%)rationalG>Fap-Fh`l6#-%'ExpandG6#*&Fa pF_oFjelFht-FV6$FjelFadlC%>8+-F[x6#-Fh`l6#-Fbfl6#*$FjelFht@$0FiflFdx>Fap*&FapF_ oFiflF_o>Fap-Fh`l6#-FbflF_y@$Fhu-Fh_l6$Fj_lFb_l>Fap*&FapF_o-%)icontentGF_yFht>F ifl-F[x6#-%%signG6#Fjel@$Fbgl>FapFdgl>Fap-Fg^l6%-Ffp6$/FipFFFapFFFax@$54-F?6$Fa pF`vFhu>F`q-FadlFaqF]q@$F]`lC.>Fd\l-FL6$Fap.%)functionG>Fd\l-%$mapG6$R6#F&F7F7F 7@$/-F[p6$FbalFNFadlFNF7F7F7Fd\l>Fgn-%&minusG6$-Fhjl6$F[y-FY6%F?-FLFc]lFB<#Ffel @$2FbalFb]lC$>Fd`lFddl@&FfalC%>FapFd`l>Fifl-F[fl6$FjelFgn@&F^elF_el-FV6$FiflFad l>Fap-Fh`l6#-Fbfl6#*&FapF_oFiflFht-FV6$Fh[mFadlC&>FiflFh[m>Fifl-F[x6#-Fh`l6#-Fb fl6#*$FiflFht@$Fbgl>FapFdgl>FapFfgl@$FhuFjgl>Fap-%)primpartGFd_l>Fifl-F[x6#-Fdh l6#Fh[m@$Fbgl>FapFdgl@$-F?6$FapF^_l>Fap-Fg^l6%FapFipFax>FapF[il@$F_il>F`qFdilF] q@$F_clC%>Fap-F[x6#Fd]m@$FhuFjgl@$0FapFdxC(>Fap-Fg^lFd_l>8.Fahl@$/Fb_mFht>Fap,$ FapFht>FapF[il@$F_il>F`qFdilF]qFgxF7F7F7%*protectedG, I0RootOf/selectorR6$F%F36#F.6#%fnCopyright~(c)~1997~Waterloo~Maple~Inc.~All~rig hts~reserved.GF7C%>FapFju@&F]t@&/Fdt.%&indexGC$@$-F?6$Fft-%$AndG6$F\u-%$NotG6#% (integerG-FR6#%Fa~constant~index~should~be~an~integerG>FapFfz0Fdt.%&labelG-FR6# %inleft-hand~side~of~the~selector~must~be~the~name~index~or~labelG45-F?6$FNF\u- F?6$FN%&rangeG-FR6#%fnselector~must~be~a~constant,~a~numeric~range~or~an~equati onG@$Fbu>FAFapF7F7F76", I0RootOf/chkindexR6$F%%"nGF7F_`mF7@$3-F?6$FNFbam52FNF_o2FAFN-FR6$%Mindex~should ~be~a~positive~integer~less~thanG,&FAF_oF_oF_oF7F7F76" M6R0 I.comparemoduleR6%%&expr1G%&expr2G%/justdiffneededG6&%#s1G%#s2G%$resG%%realG6#% ECopyright~Bruno~Salvy,~INRIA,~FranceG6"C%@$343-%(hastypeG6$9$%'RootOfG-F66$9%F 9/-%)zerotestG6$F8F<%%trueG-%'RETURNG6#%"=G@'0F<""!C)>8'FA@$-%%typeG6$F8-%)spec funcG6$%)anythingG%$absG-FC6#-9!6$-%#opG6#F8F<@$-FN6$F8$7$F8FHC$>F_p- FeoFfn>FK%&falseG@%-FN6$F8%7$FFjp-FeoF_o>FKFep@%FK>8&-%+comparexprG6 $*$&F_pFho""#*$&FjpFhoFhqC%@$/-Fen6$FHF_p%.evalc/unsplitGC$>F_p-Feo6#-%&evalfGF fn@$F]r-%&ERRORG6%%2Unable~to~compareG-.FTFfn-F]sF_o@$/-Fen6$FHFjpF`rC$>Fjp-Feo 6#-FfrF_o@$F`sFhr>Fbq-Fdq6$-%%sqrtG6#,&*$-Fen6$FioF_pFhqFio*$-Fen6$FhqF_pFhqFio -F]t6#,&*$-Fen6$FioFjpFhqFio*$-Fen6$FhqFjpFhqFio0-F?FfnFA>Fbq%">G>FbqFE@%/Fbq%# <>G@%529#""$49&-Fir6&F[s-FTFfn%$andG-FTF_oFfuFbqF/F/F/F/ M6R0 I%gdevR6$'%$fctG%*algebraicG'%%gvarG<$%%nameG/F+F'6'%#ptG%$ordG%$varG%)rightdir G%$resG6#%ECopyright~Bruno~Salvy,~INRIA,~FranceG6"C,@%-%%typeG6$&9"6#""#F,C$>8& -%#opG6$"""F;>8$-FC6$F>F;C$>FAF;>FG""!@$33-%$hasG6$FG%)infinityG0FGFT0FG,$FT!"" -%&ERRORG6$%4Invalid~limit~pointGF;>8'%%trueG@$1""$9#@(-F96$&F<6#F\o%*nonnegint G>8%Fao/Fao.%(inverseG@$3FUFV>Fhn%&falseG0Fao.%)straightG-FZ6$%1Invalid~argumen tGFao@$/F]o""%@(3-F96$&F<6#FepFco4-%)assignedG6#Feo>FeoFjp/FjpFgo@$FU>FhnF\p0Fj pF^p-FZ6$FbpFjp@$F\q>FeoFE@$/-%#isG6#2"$+"%&_XasyG%%FAILG-%-additionallyGF]r>%0 _EnvXasyintegerGF\p@)/FGFTC$>Fer-F\r6$FA%(integerG>8(-%*traperrorG6#-%*dev/prin tG6%-%(dev/devG6%-%%subsG6$/FAF`r9$FeoFeoFAFeo/FGFWC$>FerFjr>F^s-F`s6#-Fis6$/FA ,$FAFX-Fcs6%-Ffs6%-Fis6$/FA,$F`rFXF\tFeoFeoFAFeo0-%)zerotestG6#FGFin@%Fhn>F^s-F `s6#-Fis6$/FA*$,&FEFE*&FAFEFGFXFXFX-Fcs6%-Ffs6%-Fis6$/FA*&FGFE,&FEFE*$F`rFXFXFE F\tFeoFeoFAFeo>F^s-F`s6#-Fis6$/FA*$,&F\vFEFXFEFX-Fcs6%-Ffs6%-Fis6$/FA*&FGFE,&FE FEFfvFEFEF\tFeoFeoFAFeo@%Fhn>F^s-F`s6#-Fis6$/FA*$FAFX-Fcs6%-Ffs6%-Fis6$/FAFfvF\ tFeoFeoFAFeo>F^s-F`s6#-Fis6$/FA,$F_xFX-Fcs6%-Ffs6%-Fis6$/FA,$FfvFXF\tFeoFeoFAFe o@'5/F^s%*lasterrorG/F^sFar>-F"6#F<-.F"F]z-F96$F^s%*undefinedGFbzF^sF56#FerF5F5 M6R0 I.numericRootOfR6$%"pG%$valG6%%"kG%"lG%"jG6#%ECopyright~Bruno~Salvy,~INRIA,~Fra nceG6"@%54-%%typeG6$9$%(polynomG-%(irreducG6#F4-%'RootOfG6$F49%C$>8$7#-%$seqG6$ -%#opG6$"""8%/FH-FE6$""#-%(factorsGF8@%2FG-%%nopsG6#F?C%>FH-%$mapG6$%$absG-FW6$ %&evalcG-FW6$%&evalfG-%%subsG6$/%#_ZGF8$9$/F8""#C$@$333 3-%%typeG6$7$F<9%7$%)functionGFK/-%#opG6$""!F<-FN6$FPFI/-%%nopsG6#FF;,&FF;%(numericG@%2FPF`p-Fin6# -F"6#-FN6$F9F;Fgo-FF6$F;%"*GC$?&8%F;F[oC$-F"6#F`q@&/%"%GF[oFhn/Ffq%%FAILG-Fin6# FhqFgo-FF6$F;FK@./-FN6$FPF;%$expGFgo/F_r%#lnG-Fin6#-F"6#,&FipF9F^oF9/F_r%$absG- Fin6#-F"6#-FN6#F;/F_r%"OGFgo/F_r.%"LGF[s/F_r.%"QGFgo>8&-%&minusG6$-%'indetsG6$F ;Ffo<#%*constantsG@%0Fjs<"C$>%'DigitsG,&FhtF9""&F9?(8(F9F9""$F[oC%@'/F\uF9>8'-% *traperrorG6#-%%subsG6$-%$seqG6$/-FN6$F`qFjs-%&evalfG6#)#"%*y""%#\"*$F`qF^o/F`q ;F9-FU6#FjsF;/F\uF>>Fbu-Fdu6#-Fgu6$-Fju6$/F]v-F`v6#)#FevFdvF`qFgvF;>Fbu-Fdu6#-F gu6$-Fju6$/F]v,$F_vF^oFgvF;>Fbu-%-zerotest/cstG6#Fbu@%0FbuFhq-FinFdx\C$>Fbu-Fcx F`s@$FfxFgx@$-FF6$F;-%(ratpolyG6#Fdo-Fin6#-%&evalbG6#/-%'normalGF`sFP>8*-%(conv ertG6$F;%'RootOfG@$-FF6$F\z%'algnumGC$>Fbu-Fdu6#-%&evalaG6#-%'NormalG6#F\z@&0Fb u%*lasterrorG-Fin6#-Ffy6#/FbuFP-FF6$7#Fbu7$.%'symbolG%$setGC%>Fbu-FN6$F>Fi[l?&F `qFbuF[oC$@$4-%$hasG6$F\z-FN6$F9F`q-%&ERRORG6&%@bug~in~evala@Normal~with~input: GF\z%Ioutput~is~reducible~RootOf~detected~withGFi\l@$/-F"6#-Fgu6$F`qF\zF[oFhnFg o@$-Fg\l6$F;<$FdsFgs-Fin6#-F"6#-%%evalG6#-Fgu6$7$/FdsR6#%"tGF4F4F4-%$logG6#*$,& F9F9FFbu-%'testeqGF`s@$FfxFgx@&3/FjsFe t4-Fg\l6$F;^#F9C$-%)userinfoG6&F]u.%+equivalentG%9Warning:~assumption~madeGF`oF hn-Fg\lF_zC$>Fjs-F_tF_z@$/FivF9-Fin6#-F"6$-Fgu6$/-FNFjv8)-%&numerGF`s-Fgu6$/%#_ ZGF_al-FN6$F9F^alFiqF4F4F4F4, I-zerotest/cstR6#F+6#%)floatvalGF4F4C%>FhtFit>F;-Fdu6#-%2zerotest/dagevalfGFV@% -FF6$F;-%(complexG6#FbpC$>Fht,&FhtF9!"&F9-Ffy6#2-FjrF`s-%&FloatG6$F9,$FhtF^oFhq F4F4F4F4, I2zerotest/dagevalfR6#%"fG6$F,%$fctG6#%)rememberGF4@+5-FF6$F<<$FfoFbp-FF6$F<-Fh bl6#Fbdl-F`vFV-FF6$F<<$F]q%"+G-%$mapG6$FcblF<-FF6$FF<-FF6$FF`qFM@'/F`qF`zFgdlFS-F`v6#-F`q6#-Fcbl6#Ffn-F`v6#-F`q6$Ffel- Fju6$-Fcbl6#-FN6$F;FFT-F\]l6#Q$bugF4F4F4F4F4 M6R0 I&shakeR6#%%exprG6$%%amplG%$resG6#%aoCopyright~(c)~1991~by~the~University~of~Wa terloo.~All~rights~reserved.G6"C$@%33/9#""#-%%typeG6$&9"6#F2%(integerG2""!F6>8$ F6>F=%'DigitsG@'/-%&minusG6$-%'indetsG6$9$%%nameG-FC6$<#%*constantsG<#^#"""<"-% ,evalr/shakeG6$FHF=-F4FG-.%)INTERVALG6#;,$%)infinityG!""FfnC%>8%-%,evalr/evalrG 6$-%$mapG6%F"FHF=%&falseG-%'subsopG6$/""%%%NULLG-%#opG6#F\oFjnF+F+F+%*protected G M6R0 I'glimitR6$'%$fctG%*algebraicG'%%gvarG<$%%nameG/F+F'6)%#ptG%$varG%)rightdirG%$r esG%"iG%$sgnG%%coefG6#%ECopyright~Bruno~Salvy,~INRIA,~FranceG6"C+@%-%%typeG6$&9 "6#""#F,C$>8%-%#opG6$"""F=>8$-FE6$F@F=C$>FCF=>FI""!@$33-%$hasG6$FI%)infinityG0F IFV0FI,$FV!""-%&ERRORG6$%4Invalid~limit~pointGF=>8&%%trueG@$1""$9#@&/&F>6#F^o.% (inverseG@$3FWFX>Fjn%&falseG0Fbo.%)straightG-Ffn6$%1Invalid~argumentGFbo@$/-%#i sG6#2"$+"%&_XasyG%%FAILG-%-additionallyGFdp>%0_EnvXasyintegerGFio@)/FIFVC$>F\q- Fcp6$FC%(integerG>8'-%*traperrorG6#-%(dev/devG6%-%%subsG6$/FCFgp9$FGFG/FIFYC$>F \qFaq>Feq-Fgq6#-Fjq6%-F]r6$/FC,$FgpFZF`rFGFG0-%)zerotestG6#FIF[o@%Fjn>Feq-Fgq6# -Fjq6%-F]r6$/FC*&FIFG,&FGFG*$FgpFZFZFGF`rFGFG>Feq-Fgq6#-Fjq6%-F]r6$/FC*&FIFG,&F GFGF\tFGFGF`rFGFG@%Fjn>Feq-Fgq6#-Fjq6%-F]r6$/FCF\tF`rFGFG>Feq-Fgq6#-Fjq6%-F]r6$ /FC,$F\tFZF`rFGFG@&5/Feq%*lasterrorG/FeqFhpC$>-F"6#F>-.F"Fav-%'RETURNG6#%"%G-F; 6$Feq%*undefinedG-Fev6#Fjv?(F7FGFGF7F[o@)-F;6$Feq%%listGC$>8)-%&evalrG6#-%'Sign umG6#&FeqFco@)/FdwFG-Fev6#FO/FdwFZC&>8*-%+dev/lcoeffG6#Feq?&8(-%'indetsG6$Fcx%) functionGF[o@$5/-FE6$FOFhx%$sinG/F`y%$cosG>Fcx-F]r6$/Fhx-.%)INTERVALG6#;FZFGFcx >Fdw-Ffw6#-Fiw6#Fcx@'F]x-Fev6#FVF`x-Fev6#FYC$>F`vFbv-Fev6#Fbv/FdwFO@%-F;6$-FE6$ F@FeqFaw>Feq7%-FE6#F`[lFOFV>FeqF`[lC$>F`vFbvFjz4-FT6$Feq%)_XasytabG-FevFfx45-FT 6$FeqFby-FT6$FeqFdyF]\lC$?&Fhx-Fjx6$Feq<$-Fby6#%)anythingG-FdyFj\lF[o>Feq-F]r6$ FhyFeq-Fev6#-FfwFfxF7F7F7F7 M6R0 I+dev/divideR6%%%dev1G%%dev2G%"pG6#%$resG6#%RCopyright~Bruno~Salvy,~INRIA~Rocqu encourt,~FranceG6"C%>%0_EnvXasyintegerG%%trueG@%4-%$hasG6$9$%-_saddlepointG>8$- %(dev/prdG6$F6-%(dev/powG6%9%!""9&>F97%F7&F66#""#-F;6$&F66#""$-F>6%&F@FKFAFB-%' RETURNG6#F9F,F,F,F, M6R0 I+dev/reduceR6$%$devG%"nG6&%"iG%%restG%'resultG%"lG6#%RCopyright~Bruno~Salvy,~I NRIA~Rocquencourt,~FranceG6"@)4-%%typeG6$9$%%listGF44-%(hastypeG6$<#-%#opG6#F4F 57#-F<6$;"""-%$minG6$,&9%""#""$FB-%%nopsGF=F41-%+dev/lengthGF=,&FGFBFBFBF4C%>8% FO?(8$FHFHFJ0FR""!@%4-F26$&F46#FTF5C$>&8&Ffn6$Fen&F46#,&FTFBFBFB>FR,&FRFB!""FBC $>8'-FN6#Fen@%1FdoFRC$>FinF[o>FR,&FRFBFdoFaoC$>Fin6$-F"6$FenF`oF\o>FRFV@'35/FTF H3/FT""%/FRFV/&&Fjn6#FHF^qFV&F]q6#FBFjp7$&F4F`q-%$seqG6$&Fjn6#,$FTFH/FT;FB,&-%% iquoG6$FTFHFBFaoFB7$Fbq-Fdq6$Ffq/FT;FBF\rF.F.F.F. M6R0 I'dev/lnR6$%"uG%"nG6)%$degG%%exprG%#nuG%&rootsG%%initG%'newvarG%"xG6$%)remember G%RCopyright~Bruno~Salvy,~INRIA~Rocquencourt,~FranceG6"@'-%%typeG6$9$%*undefine dGF84-F56$F7%%listG@)-%)zerotestG6#F7-%&ERRORG6#%&ln(0)G-F?6#,&F7"""!""FH""!3%0 _EnvXasyintegerG/-%$mapG6$%#opG-%'indetsG6$F7&%)_XasytabG6#%)anythingG<#FHC&>8% -%'expandG6#-%%subsG6$/&FV6#FH*$8*FIF7>%1_EnvAllSolutionsG%%trueG>8'<#-%&solveG 6$FfnFao@%3-F56$Ffo-%$setG6#%%nameG-%#isG6$-FQ6#Ffo%(integerGFA-%#lnGF@FipC$>8& -%%nopsGF@@'3/F]q""&/&F76#Fcq%)infinityGC%>8$&F76#""$@%-F56$&F76#""#F<>8(-F"6$7 %-FQ6#FarFJFgq9%>Fer-F"Fjr@%/FjqFJFer@%-F56$&F7F_oFhp@%-%)assignedG6#&%(_equivX G6#,&FcsFHFHFH-%(dev/addG6$Fer7'F[t,$FjqFIFIFJFgqC$>Fhs*$-Fjp6#*$&Fis6#FcsFIFIF \tC$>8)-%-dev/indexifyG6#-Fjp6#&FVFht-F]t6$Fer-%'subsopG6$/Fcr-%/dev/multbyreal G6$&F[uFbrFjqF[u/F]qF]rC$>Fer-F"6$7'FcsFarF[rFJFgqF[s@%3-F56$FerF8$*$,&FDFDF;FEFE-FK6$-F"6$-FK6$FMFVFSFV-FH6%-%(dev/addG6$-FK6$F1FEFDFEFS C$>8%-%&evalrG6#-%'SignumG6#F0@'/FboFEFin/Fbo%%FAILG-%&ERRORG6#F\p-%-dev/endofd evG6%F1,&FSFDFDFD-%$mapG6$%#opG7#-%$seqG6$7$FD8&/F]q;F4FcpF-F-F-F- M6R0 I(dev/prdR6$%"UG%"VG64%$degG%"iG%"jG%"uG%"vG%*candidateG%%deg1G%'exactuG%'exact vG%'inter1G%#nuG%#nvG%'resultG%"lG%&countG%(tomult1G%(tomult2G%$sigG6$%)remembe rG%RCopyright~Bruno~Salvy,~INRIA~Rocquencourt,~FranceG6"@)5-%%typeG6$9$%*undefi nedG-FA6$9%FDFD4-FA6$FC%%listG@%4-FA6$FGFK*&FC"""FGFQ-%/dev/multbyrealG6$FGFCFM -FS6$FCFGC+@'/-%#opG6$FQFC-Fen6$FQFGC$>8'FC>8(FG2FZFgnC$>F[oFC@%/&FG6#-%%nopsG6 #FG%)infinityG>F]o7'FZ7#-Fen6$;FQ,&FeoFQ!"#FQFG""!FapFho>F]o7%FZFGFapC$>F]oFG@% /&FC6#-Ffo6#FCFho>F[o7'Fgn7#-Fen6$;FQ,&FjpFQF`pFQFCFapFapFho>F[o7%FgnFCFap>8.-F fo6#F[o>8/-Ffo6#F]o>8+-%&evalbG6#/&F[o6#FfqFho>8,-F`r6#/&F]o6#FjqFho>8$7#7%,&&F [o6#""$FQ&F]oFbsFQ""#Fes>&806#FQ&F[oFis?(82FesFQF=%%trueGC(>8*&F]sFis>8-Fap?(8% FesFes-Ffo6#F`tF]tC%@%5/&F`t6#Fet,&FfqFQ!""FQ4-FA6$&F[o6#F\uFK>83Fcu>Ffu7%-Fen6 #FcuFapFho@%5/&F`t6#,&FetFQFQFQ,&FjqFQF_uFQ4-FA6$&F]o6#F^vFK>84Fev>Fhv7%-Fen6#F evFapFho>Fct-%(dev/addG6$Fct-F"6$FfuFhv@$3-FA6$FctFK/&Fct6#-Ffo6#FctFho>Fct7#-F en6$;FQ,&FjwFQF`pFQFct@%0FctFap>&Fhs6#F\t6$Fct&F`tFis>Fex%%NULLG@%534F^r-%'memb erG6$F^u7#-%$seqG6$&F`t6#,$FetFes/Fet;FQ-%%iquoG6$FftFes34Ffr-F`y6$Fav7#-Fdy6$& F`t6#,&FetFesFQFQFiyC$@$/FctFap>Fex6$FQFhx[C&>F]s-%'subsopG6$/FQFjxF]s?(FetFesF esFftF]t@$30F\uF^u0&F[o6#,&F\uFQFcsFQFhoC&>8),&Fi[lFQ&F]o6#,&F^vFQFQFQFQ>85FQ?( 8&FQFQ-Ffo6#F]s/Fd\lFQC$>Fd\l-%&evalrG6#-%'SignumG6#,&F^\lFQ&&F]s6#Ff\lFisF_u@( /Fd\lF_u>F]s7%-Fen6$;FQ,&Ff\lFQF_uFQF]s7%F^\l,&F\uFQFesFQF^v-Fen6$;Ff\lFg\lF]s/ Fd\lFap@$4-F`y6$7$F_^lF^v7#-Fdy6$7$&Fd]l6#,$81Fes&Fd]l6#,&F`_lFesFQFQ/F`_l;FQ-F \z6$-Ffo6#Fd]lFes>F]s-Fa[l6$/Ff\l7%-FenFi_lF_^lF^vF]s/Fd\l%%FAILG-%&ERRORG6#Fa` l@$Fi\l>F]s7$-FenFh\lF^^l?(FetFesFesFftF]t@$30F^vFav0&F]o6#,&F^vFQFcsFQFhoC&>F^ \l,&&F[o6#,&F\uFQFQFQFQF^alFQ>Fd\lFQ?(Ff\lFQFQFg\lFi\lC$>Fd\lF\]l@(Fg]l>F]s7%Fj ]l7%F^\lF\u,&F^vFQFesFQF`^lFc^l@$4-F`y6$7$F\uF_blFi^l>F]s-Fa[l6$/Ff\l7%F_`lF\uF _blF]sF``lFb`l@$Fi\l>F]s7$Fh`lF^bl@$/Fg\lFapF][l@'3/&&Fhs6#FesFdclFap52F\tFcs3/ F\tFcs/&&FhsFbsFdclFho@%Ffcl&FcclFis7%-Fen6#F]dlFapFhoF^cl7%-Fdy6$&FhsF]u/Fet;F QF\tFapFho7#FbdlF=F=F=F= M6R0 I,dev/instancR6#%%exprG6&%"iG%$resG%$limG%$varG6#%RCopyright~Bruno~Salvy,~INRIA ~Rocquencourt,~FranceG6"@%4-%%typeG6$9$%%listG>8%F3C&@%0&F36#-%%nopsG6#F3%)infi nityG>8&F<>FA,&F<"""!"#FD>F6""!>8'&%)_XasytabG6#&F36#FD?(8$""#FQFA%%trueG@%-F16 $&F36#FPF4>F6,&F6FD*&-F"6#FVFD)FI&F36#,&FPFDFDFDFDFD>F6,&F6FD*&FVFDFgnFDFDF-F-F -F- M6R0 I(dev/absR6$%$devG%"nG6$%'impartG%$sigG6#%RCopyright~Bruno~Salvy,~INRIA~Rocquen court,~FranceG6"@'-%%typeG6$9$%*undefinedGF24-F/6$F1%%listG-%&evalcG6#-%$absG6# F1C$>8$-%+dev/impartGF<@%4-%$hasG6$F?^#"""C$>8%-%&evalrG6#-%'SignumG6#-%+dev/lc oeffGF<@)/FKFHF1/FK!""-%/dev/multbyrealG6$F1FW/FK""!7#Ffn%%FAILG-%(dev/powG6$-% (dev/addG6$-Fjn6%-%-dev/realpartGF<""#9%-Fjn6%-%%subsG6$/FGFHF?FcoFdo#FHFcoF,F, F,F, M6R0 I&dev/LR6$%"uG%"nG6%%"iG%%initG%$sigG6#%RCopyright~Bruno~Salvy,~INRIA~Rocquenco urt,~FranceG6"@%/&9$6#""$""!@'34-%%typeG6$&F16#""#%%listG0-%)zerotestG6#,&F;""" !""FD%&falseG-%/dev/multbyrealG6$-%'dev/lnG6$-FH6$-%'subsopG6%/F=%%NULLG/F3FSF1 FE9%FEF7C$>8%*$,&FDFDF;FEFE-%(dev/addG6$-%#lnG6#FX-F"6$-FH6$FOFXFU-FH6$-FK6$-Ff n6$-FH6$F1FEFDFUFEC$>8&-%&evalrG6#-%'SignumG6#F0@'/FioFEF_o/Fio%%FAILG-%&ERRORG 6#Fcp-%-dev/endofdevG6%F1,&FUFDFDFD-%$mapG6$%#opG7#-%$seqG6$7$*$8$FEFeq/Feq;FD, &FUFDF=FDF-F-F-F- M6R0 I)dev/signR6#%$devG6"6#%RCopyright~Bruno~Salvy,~INRIA~Rocquencourt,~FranceGF&@% 4-%%typeG6$9$%%listG-%&evalrG6#-%'SignumG6#F.&F"6#&F.6#""#F&F&F&F& M6R0 I(dev/powR6%%"uG%"pG%"nG6-%%exprG%"iG%"xG%%res1G%$resG%$indG%&interG%'binpowG%" qG%%lvarG%&rootsG6$%)rememberG%RCopyright~Bruno~Salvy,~INRIA~Rocquencourt,~Fran ceG6"@+-%%typeG6$9$%*undefinedGF=4-F:6$F<%%listG@)33-F:6$F<%'algnumG-%$hasG6$F< %'RootOfG-F:6$9%%(integerG-%&evalaG6#-%'NormalG6#)F8)-%&minusG6$-%'indetsG6$F<%%nameG<#%*constantsG>8--F_o6$F ]o-Fbo6$F<&%)_XasytabG6#%)anythingG@$/-%%nopsG6#FhoFgn>8'-%&evalcG6#-%$absG6#F< ?&8%-Fbo6$Fgp<$-%)specfuncG6$F`p%$sinG-Fdq6$F`p%$cosG%%trueG>Fgp-%%subsG6$/F_q- .%)INTERVALG6#;!"""""Fgp>Fgp-%*traperrorG6#-%&evalrG6#Fgp@)330Fgp%*lasterrorG-F :6$FgpFA/-F\s6#-%'SignumG6#-%#opG6$Ffr-F\tF]sFfrFV3/%0_EnvXasyintegerGFjq/-F_o6 $-Fbo6$F]oFdoFeo<#&F^p6#FfrC&>8$-F]r6$/Fht*$8&FerF<>%1_EnvAllSolutionsGFjq>8.7# -%&solveG6$F\uFau@%3-F:6$Feu-%$setG6#Fdo-Fin6$-F\t6#FeuFOF=FV/FgpFbs-%&ERRORGF] sFV5/-FdpF]q""$/&F<6#""&%)infinityG@'-F:6$&F<6#""#FAC$>8*-F"6%7%-F\t6#FdwFgnF`w FN9&@'0&Fiw6#-Fdp6#FiwF`w7%&F8,FN>8(Ffr>8+FF_z-%( dev/prdG6$F_zFaz>Faz-%+dev/reduceG6$-F\[l6$FazFazF_x>F]z-%%iquoGFizF[[l/FNFgnFf r-F"6%-F"6%F<,$FNFerF_xFerF_x-F\[l6%-F"6%7'FgxFdwFixFgnF`wFNF_x-%1dev/pow/endof devG6%-F\[l6%-F"6%Fa\lFerF_x7$Fgx-F\t6$;""%FjvFF_q,$*&FNFfr,& FNFfrFerFfrFfr#FfrFfw?(FauF[wFfrF_xFjqC$>F_q*(F_qFfr,(FNFfrFauFerFfrFfrFfrFauFe r>&F\u6#Fau6$F_qFau-%-dev/endofdevG6%F&8%6#F>6$-F"6$&F4FGF/&F46#,&F>F;F;F;7$&F46#F;- %$seqG6$&FF6#,$F>F?/F>;F;-%%iquoG6$F@F?F,F,F,F, M6R0 I+dev/lengthR6#%$devG6#%"iG6#%RCopyright~Bruno~Salvy,~INRIA~Rocquencourt,~Franc eG6"@%4-%%typeG6$9$%%listG"""-%$addG6$-F"6#&F06#,$8$""#/F;;F2-%%iquoG6$-%%nopsG 6#F0F8$-F[o6$-%(dev/powG6%F4*$FgnFQF^oF^o7$FG-F?6$-FB6#7$-FB6$,$F EFFFgo*&-FB6$FDFgoFGFgnFQ/FE;FG-FL6#-FO6$FgoFFF,F,F,F,, I0dev/invert/doitRF$6+%"xG%"oG%#ndGF)%"kG%"pG%$ppvG%#pvG%"vGF*F,C+>8&FN@%0&F46# F]r%)infinityG>FE-%$minG6$F^oF`r>FEF^o@$3/F`rFbr/F]r""&-%'RETURNG6#7'FG*$-FB6$F FF4FQFG""!Fbr>8,*&FgoFGFbsFQ>8)-%(convertG6$7#-F?6$*&-FB6$,$8'FFF4FG)Fgo-FB6$,& FdtFFFGFGF4FG/Fdt;FG,&-FL6$F]rFFFGFQFG%"+G>8(FG?(F,FGFGF,1,&F`uFFFGFGFEC&>8+-%6 dev/invert/powcomposeG6&FisFfsFgoFcu>8*-%:dev/invert/pprimeknowingpG6&FfuFfsFgo Fcu>Ffs,&FfsFG-%5dev/invert/powdivideG6&,&FfuFGFgoFQF[vFgoFcuFQ>F`uFcu@$2F`uFEC %>Ffu-Fhu6&FisFfsFgoFE>F[v-F]v6&FfuFfsFgoFE>Ffs,&FfsFG-Fbv6&FdvF[vFgoFEFQ-%,dev /cleanupG6#7&FV-F?6$-FB6#7$-%&coeffG6%FfsFgoF`uF`u/F`u;Fds,&FEFGFQFGFGFEF,F,F,F ,, I6dev/invert/powcomposeR6&%"QG%"PGFcqF&6+%"mG%#pmG%#prG%$pr1G%"lGF)%"sGFgq%"qGF *F,C,>Ffu-%7dev/invert/powtruncateG6%F^o9&9'>Ffs-Fby6%F4FdyFey>Fgo-%&isqrtG6#-% &truncG6#,$*&FeyFG-%'lengthG6#FeyFQ$"*4G>K$!")>FE-Fby6%FfuFdyFgo>F]rFG>Fdt,&Ffu FGFEFQ>F`u-%&F[v6#FdsF`u?(FisFGFG,&- FL6$FeyFgoFGFGFG%%trueGC%>F`u-F]v6&F`uFEFdy,&FeyFGFisFQ>F]r-Fby6%-%'expandG6#*& F]rFGFdtFGFdyFey>&F[v6#Fis*&-Fby6%-Fc\l6#*&F`uFGF]rFGFdyFeyFG-%*factorialGFh\lF Q-F^s6#-F[t6$7#-F?6$Fg\l/Fis;FdsFf[lF^uF,F,F,F,, IFE-%'degreeG6 $F4Fdy>FgoFG?(F,FGFGF,2FgoFE>Fgo,$FgoFF>Fgo,$Fgo#FGFF>&F]r6#FGFay>FdtFF?(F,FGFG F,1FdtFgoC$>&F]r6#Fdt-Fby6%-Fc\l6#*$&F]r6#,$FdtF\_lFFFdyFey>FdtFct-%@dev/invert /powcomposesimpledoitG6'F4FgoF]rFdyFeyF,F,F,F,, I5dev/invert/powdivideR6&FgqF^yFcqF&6&%"uG%#bkGF)F^^lF*F,C'@$/-%%subsG6$/FdyFds F^oFds-FY6#%0not~implementedG>FE7#-F?6$-F]x6%F^oFdyF]r/F]r;FdsFey>&FgoFd[l*&-F] x6%F4FdyFdsFG&FEF__lFQ?(F]rFGFGFeyFi[l>&FgoFar*&,&-F]x6%F4FdyF]rFG-F[t6$7#-F?6$ *&&FgoFf_lFG&FE6#,(F]rFGFdtFQFGFGFG/Fdt;Fds,&F]rFGFQFGF^uFQFGF`blFQ-F[t6$7#-F?6 $*&FcblFG)FdyF]rFGFialF^uF,F,F,F,, I6dev/invert/powinverseR6%F^yFcqF&Ff`lF*F,C'@$/-F]al6$/F^oFdsF4FdsF`al>FE7#-F?6 $-F]x6%F4F^oF]r/F]r;FdsFdy>F\bl*$F`blFQ?(F]rFGFGFdyFi[l>Fcbl,$*&FhblFGF\blFGFQ- F[t6$7#-F?6$*&F^clFG)F^oFdtFG/FdtF\elF^uF,F,F,F,, I@dev/invert/powcomposesimpledoitR6'F^yF]yF_^lFcqF&6$%#q1G%#q2GF*F,C&@$/F^oF\_l -F^s6#-F]al6$/Fey&FdyF__lF4>Fgo-Fby6%F4Fey,&F^oFGFQFG>FE-%'normalG6#*&,&F4FGFgo FQFG)FeyF^oFQ-Fby6%,&-Fa`l6'Fgo,$F^oF\_lFdyFey9(FG-Fc\l6#*&&Fdy6#F^oFG-Fa`l6'FE FjglFdyFey,&F[hlFGF^oFQFGFGFeyF[hlF,F,F,F,, I7dev/invert/powtruncateR6%%$polGFcqF&6#F)F*F,C%@$1-Fc^l6$F4F^oFdy-F^sFP@$2Fdy- %(ldegreeGF]il-F^sFd[l-F[t6$7#-F?6$*&-F]x6%F4F^oFgoFG)F^oFgoFG/FgoF\elF^uF,F,F, F,, I:dev/invert/pprimeknowingpR6&%%poffG%"fGFcqF&F,F*F,-Fbv6&-%%diffGFd^l-Ffjl6$F^ oFdyFdyFeyF,F,F,F, M6R0 I+dev/impartR6#%%exprG6&%"iG%'resultG%"nG%$auxG6#%RCopyright~Bruno~Salvy,~INRIA ~Rocquencourt,~FranceG6"@%4-%%typeG6$9$%%listG@%-F16$F3%)realconsG""!*&^#"""F<- %&evalcG6#-%#ImG6#F3F8&-%%nopsGFB?(8$""#FJ,&FEF8'-F"6#7%-%#opG6#FRF9%)infinityG@$3-F16$FVF4/&FV6#-FG6#FVFgn>FV-%'subsopG 6%/F_o%%NULLG/,&F_oFFV-F"Ffn@%0FVF9>&8%FS6$FV&F36#,&FIFF_pFfo >F`p-%$seqG6$&F`p6#,$FIFJ/FI;F<-%%iquoG6$FKFJ@'/&F36#FEFgn@'/F`pFfoF93/-FG6#7#F `pFJ/&F`p6#FJF9&F`p6#F<7&&F3FarF`pF9Fgn-F16$&F36#,&FEF8&-%%nopsGF>@$30&F66#FA%)infinityG0-%&evalrG6#-%'Signu mG6#FG"""-%&ERRORG6#%;Not~a~convergent~expansionG>8'FQ?(8$""$""#FA/-FL6#-FO6#&F 66#FY!""@'3/F[oF]o4-F46$&F66#,&FYFQF]oFQF;C$@$4-%)assignedG6#&%(_equivXG6#,&&F6 6#FQFQF]oFQ>F]p*$-F=6#*$&F^p6#FapF]oF]o>FW-%(dev/prdG6$FW7'F`p-F=6#*&-%,dev/ins tancG6#-%+dev/impartG6#FdoFQ&%)_XasytabGFipF]o,$-%-dev/realpartGFgqF]o""!FIFao> FW-F\q6$FW-%/dev/multbyrealG6$-%(dev/powG6%-%-dev/indexifyG6#-.F=6#7%FapFQF[oF[ r9%-F=6#-Fcq6#7'FapFeqF[oF]rFI?(8(FZFen-FCFgq%%trueG>FW-F\q6$FW-Fbr6$-Fhr6#-F[s 6#-F\r6#7'Fap7$-%#opG6$FQFdo-Fft6$;,&FesFQF]oFQFesFdoF[oF]rFI-F=6#-Fcq6#-FfqFbt @$/-Fft6$FYF6F]rC$@%4-F46$-Fft6$FfoF6F;>FW-Fbr6$FW-F"6$FjuF^s>FW-F\q6$FW-F"6$7% -Fft6#FjuF]rFIF^s>FY,&FYFQFenFQ@'2FYFAC&>8%#FQ""'>8)7*FQF]rFQFQ#FQFenFenF`wFZ?( Fes""%FQF^sFgsC$>F_w*&F_wFQFesF]o>Fcw7%-Fft6#FcwF_wFes-F\q6$FW-%-dev/endofdevG6 %7$Fap-Fft6$;FfoFAF6F^sFcw/FGFIFW-F\q6$FW7'FapFQF]rFQFGF16#F^pF1F1 M6R0 I-dev/evenpartR6#%$devG6%%"aG%"nG%"iG6#%RCopyright~Bruno~Salvy,~INRIA~Rocquenco urt,~FranceG6"@'4-%%typeG6$9$%%listGF20-%#opG6$"""F2F8-%&ERRORG6#%=a~Puiseux~se ries~is~expectedGC%>8%,&-%%iquoG6$-%%nopsG6#F2""#F8!""F8?(8&F8F8F?%%trueG@%-F06 $-F66$,&FJFGF8F8F2%%evenG>&8$6#FJ6$-F66$,$FJFGF2FO>FT%%NULLG7&F5-%$seqG6$FT/FJ; F8F?-F66$,&F?FGFGF8F2-F66$,&F?FG""$F8F2F,F,F,F, M6R0 I.dev/conjugateR6#%%exprG6&%"iG%'resultG%"nG%$auxG6#%RCopyright~Bruno~Salvy,~IN RIA~Rocquencourt,~FranceG6"@%4-%%typeG6$9$%%listG-%*conjugateG6#F3C&>8&-%%nopsG F7?(8$""#F?,&F:"""!"$FA%%trueGC$@%-F16$&F36#F>F4C$>8'-F"6#7%-%#opG6#FH""!%)infi nityG@$3-F16$FLF4/&FL6#-F<6#FLFT>FL-%'subsopG6%/Ffn%%NULLG/,&FfnFA!""FAF]oFL>FL -F"FR@%0FLFS>&8%FI6$FL&F36#,&F>FAFAFA>FfoF]o>Fgo-%$seqG6$&Fgo6#,$F>F?/F>;FA-%%i quoG6$F@F?@'/&F36#F:FT@'/FgoF]oFS3/-F<6#7#FgoF?/&Fgo6#F?FS&Fgo6#FA7&&F3FhqFgoFS FT-F16$&F36#,&F:FAF`oFAF47&FjqFgo-F"6#F]rF[q7&FjqFgoFAF[qF-F-F-F- M6R0 I,dev/cleanupR6#%$devG6%%"iG%"aG%"bG6#%RCopyright~Bruno~Salvy,~INRIA~Rocquencou rt,~FranceG6"@%4-%%typeG6$9$%%listG@%/-%)zerotestG6#F2%%trueG""!F2C&>8%-%$mapG6 $9!7#-%$seqG6$&F26#,&8$""#FJ"""/FI;F:,&-%%iquoG6$-%%nopsGF8FJFK!""FK?(FIFKFK-FS 6#F=F9@'/-%#opG6$FIF=F:>&8&6#FI%%NULLG-F06$FZ7%%)anythingGF_oF:>Fhn6$&FZ6#FJ&F2 6#,&FIFJFKFK>Fhn6$FZFdo>Fin-FD6$Fhn/FI;FKFV@%/FinF[oF:7$-Fen6$FKF2FinF,F,F,F, M6R0 I(dev/addR6$%"UG%"VG60%"iG%"jG%'resultG%"uG%"vG%#nuG%#nvG%%deguG%%degvG%&interG %$sigG%'toadd1G%'toadd2G%"sG6$%)rememberG%RCopyright~Bruno~Salvy,~INRIA~Rocquen court,~FranceG6"@+5-%%typeG6$9$%*undefinedG-F=6$9%F@F@/F?""!FC/FCFEF?45-F=6$F?% %listG-F=6$FCFKC$>8&,&F?"""FCFR@%5/-%)zerotestG6#FP%%trueG3-F=6$7$F?FC7$%&float GFin52-%$absGFX-%'subsopG6$/FRFRF?2F\o-F_o6$FaoFCFEFPC*@+4FIC$>8'7'&FC6#FRF?FEF E%)infinityG>8(FC4FLC$>FjoF?>F`p7'&F?F]pFCFEFEF^p/-%#opG6$FRF?-Fip6$FRFCC$>FjoF ?>F`pFC2FhpF[qC$>FjoF?@%/&FC6#-%%nopsG6#FCF^p>F`p7'Fhp7#-Fip6$;FR,&FgqFR!"#FRFC FEFEF^p>F`p7%FhpFCFEC$>F`pFC@%/&F?6#-Fhq6#F?F^p>Fjo7'F[q7#-Fip6$;FR,&FjrFRFarFR F?FEFEF^p>Fjo7%F[qF?FE>8)-Fhq6#Fjo>8*-Fhq6#F`p>8$""$>8%F_t?(81""#FRF931F^tFfs1F atFjsC)>8+&Fjo6#F^t>8,&F`p6#Fat@$/FjtF^pC%>&FP6#Fct-Fip6$;,&FatFR!""FRFjsF`p>Fc t,&FctFRFRFR[@$/F^uF^pC%>Feu-Fip6$;,&F^tFRF[vFRFfsFjo>FctF]vF^v>8.-%'signumG6#, &FjtFRF^uF[v@$-F=6$Fiv-%)specfuncG6$%)anythingGF[wC$>Fiv-%+comparexprG6$FjtF^u@ &/Fiv%"FivF[v/Fiv%">G>FivFR@)/FivFRC$>Feu6$&F`p6#FjuF_u>Fat,&FatFRFdtFR/FivF [vC$>Feu6$&Fjo6#FfvF[u>F^t,&F^tFRFdtFR3/Fiv%%FAILG4-F=6$7$FjtF^u7$.%)realconsGF jy-%&ERRORG6#FdyC)@%5/F^tFfs4-F=6$F^yFK>8/F^y>Fgz7%-Fip6#F^yFEF^p@%5/FatFjs4-F= 6$FfxFK>80Ffx>Fc[l7%-Fip6#FfxFEF^p>8--F"6$FgzFc[l@$33-F=6$Fi[lFK52F^tFfs2FatFjs /&Fi[l6#-Fhq6#Fi[lF^p>Fi[l7#-Fip6$;FR,&Fg\lFRFarFRFi[l@'0Fi[lFE>Feu6$Fi[lFjt5Fb zF^[l>FeuF]y>Feu%%NULLG>F^tFay>FatFix>FP7$-Fip6$FRFjo-%$seqG6$&FPF\u/F^t;Fdt,&F ctFRF[vFR@+/-FhqFXFR@%2FfsF^t7%&FjoF]p&Fjo6#,&FfsFRF[vFR&Fjo6#Ffs7%&F`pF]p&F`p6 #,&FjsFRF[vFR&F`p6#Fjs3/Ff^lF_t/&FP6#F_tF^p@%-F=6$&FP6#FdtFK7%-Fip6#F_`lFEF^pFE 3Fh_l/Fj_lFE@%F]`lF_`l7%FRF_`lFE33/Ff^l""&/&FP6#F[alF^pFe`l@%F]`lFa`lF_`lFPF9F9 F9F9 M6R0 I&dev/OR6#%$devG6"6#%RCopyright~Bruno~Salvy,~INRIA~Rocquencourt,~FranceGF&@'-%% typeG6$9$%*undefinedGF.4-F+6$F-%%listG"""7%&F-6#F3-F"6#&F-6#""#&F-6#""$F&F&F&F& M6R0 I+dev/RootOfR6$%(fctglobG%"pG6-%"fG%"xG%%solsG%$resG%"iG%$algG%%lalgG%#fcG%$fct G%"jG%"yG6#%RCopyright~Bruno~Salvy,~INRIA~Rocquencourt,~FranceG6"C$Z%>8,-%'norm alG6#9$F5>F9F=@%4-%%typeG6$F9-%'algfunG6$%)anythingG7$%&_XasyG%#_ZGC'>8+F9>8*7# -%#opG6#-%'indetsG6$F9%'RootOfG>FM-%%subsG6$<#-%$seqG6$/&FO6#8(&8)F\o/F]o;"""-% %nopsG6#FOFM>8$7#-%&solveG6$FMFJ@%4-%$hasG6$FgoFW-%$mapG6&%(dev/devG-FZ6$<#-Fhn 6$/F^oF[oF`oFgo9%F[qC)>Fgo7#-Fjo6$FMFI@$F^p-%&ERRORG6#%:cannot~expand~this~Root OfG>8'-Fdp6%-FZ6$FgpFMFboFbo@%4-F_p6$-FR6$""#FgqFJ>8&7#%)infinityG>Fdr7#-FjoF_r >Fgq7"?&F]oFdr%%trueG@'/F]oFfr>Fgq7$-FR6#Fgq-FR6#-%-dev/implicitG6%-Fdp6%-FZ6$/ FJFI-FR6$FboFgoF[qF[q7'FboFbo!""""!FfrF[q/F]oFbt>Fgq7$Fbs-FR6#-Fbp6&%(dev/powG- Fgs6%-Fdp6%-FZ6$/FJ*$FIFatF^tF[qF[qF`tF[qFatF[qC$@%-FB6$F]o-FW6#%*algebraicGC&> %'DigitsG,$F\vFbr>8--%&evalfGF\o>F\v-%%iquoG6$F\vFbr>F_v-FW6$-FR6$FboF]o-Fav6#F _v>F_vF]o>Fgq7$Fbs-FR6#-Fbp6&R6%F)F2F&F5F5F5-%/dev/multbyrealG6$-%(dev/addG6$Fa t-Fjt6%F=Fat9&F[qF5F5F5-Fgs6%-Fdp6%-FZ6$/FJ*&F_vFbo,&FboFboFbuFatFboF^tF[qF[qF` tF[q,$F_vFatF[qFgq-%2dev/RootOf/algfunG6&-%&numerG6#-%%evalG6#-FZ6$7$/FI*$8%Fat /FJ8.F9FgyFiyF[qF5F5F5F5, I2dev/RootOf/algfunR6'%$PolGF)F2%$ordG%9optional_positive_slopesG69%$polG%"aG%" uGF,F1%$ptsG%&alphaG%%miniG%$degG%&thetaG%%jminG%#slG%'lastptGF&%#puG%$nb2GF+%" rG%#nbG%"qG%#a0G%+normalizerG%#eqGF3F5C.>Fgo-%(collectG6%F=7$F^xF[q%&evalaG>F9- %'degreeG6$FgoF^x>F_o-%'selectG6$R6#F)F5F5F50&F=6#FbrFbtF5F5F57#-Fhn6$7$Fgq-%&c oeffG6%FgoF^xFgq/Fgq;FbtF9>F_o7#-Fhn6$7$&Fgq6#Fbo-%(ldegreeG6$&FgqFg\lF[q/FgqF_ o>86Fbt>80&F_oFg]l?(FgqFbrFbo-Fdo6#F_oF]sC(>FMFfr?(F]oFgqFboFc^lF]sC$>F_v*&,&&F ^oFg\lFbo&F`^lFg\lFatFbo,&&F^oFg]lFbo&F`^lFg]lFatFat@$2F_vFMC$>FMF_v>FiyF]o>F^^ l,&F^^lFboFboFbo>&FO6#F^^l,$FMFat>FgqFiy>F`^l&F_oFcs>83Fbt>FO<#-Fhn6$&FOFcs/Fgq ;FboF^^l@$/9#""&>FO-Fa\l6%FBFO.%,nonnegativeG?&8/FOF]sC(>85-%&denomG6#Fbal>81-F g[l6$-FZ6%/F[q)F[qFeal/F^x*&)F[q-F]yFhalFboF^xFboFgoF[q@$0-Fi]l6$FjalF[qFbt>Fja l-Fg[l6$*&FjalFbo)F[qFgblFatF[q>87-Fg[l6$-F]]l6%FjalF[qFbtF^x@$0-Fi]l6$F_clF^xF bt>F_cl-Fg[l6$*&F_clFbo)F^xFfclFatF^x?&Fdr&-%(sqrfreeGFgclFg\lF]sC$Z'>887#-Fj[l 6#-FW6$&FdrFg]lF^xQNreducible~RootOf~detected.~~Substitutions~areF5>Fddl-Fbp6%F Z&%.lastexceptionG6#""$FhdlF5YF5?&FddlFddlF]s@%/&FdrFg\lFboC(@%-FB6$FddlFW>89Fj [l>F]flF;>&Fgy6#FbtFddl>82Fjal?(FgqFboFbo,&9'FboFatFboF]sC%>Fcfl-Fg[l6$-FZ6$/F^ x,&&Fgy6#,&FgqFboFatFboFbo*&F[qFboF^xFboFboFcflF[q>8:-F]]l6%FcflF[qFgq@%/-F]\l6 $FdglF^xFbo>&FgyFcs-F]fl6#,$*&-F]]l6%FdglF^xFbtFbo-F]]l6%FdglF^xFboFatFat-Fcq6# %0unforecast~caseG>F``l,&F``lFboFboFbo>&846#F``l-Fbp6$FR7&Fbo-Fhn6$7$F\hl,&Fbal Fbo*&FgqFboFealFatFbo/Fgq;FbtFeflFboFffl?&Fgq-Fjx6'-FZ6$7$/F[q)F[qFgel/F^x,&Fdd lFboFbglFboFjalF[qF^xFeflFboF]sC$>F``lFihl>F[il-Fbp6$FR7%Fbo7$FddlFbal-Fhn6$7$& Fgq6#,$F]oFbr*(,&&Fgq6#,&F]oFbrFboFboFboFboFboFboFgelFatFealFat/F]o;Fbo-Fdv6$-F doFcsFbr7#-Fhn6$&F\ilFcs/Fgq;FboF``lF5F5F5F5 M6R0 I-dev/binomialR6%%"uG%"vG%"nG6"6#%RCopyright~Bruno~Salvy,~INRIA~Rocquencourt,~F ranceGF(@%5-%%typeG6$9$%*undefinedG-F.6$9%F1F1-%(dev/prdG6$-%*dev/GAMMAG6$-%(de v/addG6$F0"""9&-%(dev/powG6%-F66$-F96$-F<6$F4F>F?-F96$-F<6$F0-F<6$-%/dev/multby realG6$F4!""F>F?FRF?F(F(F(F( M6R0 I,dev/compareR6$%%Dev1G%%Dev2G6(%"iG%%dev1G%%dev2G%#n1G%#n2G%$sigG6#%RCopyright ~Bruno~Salvy,~INRIA~Rocquencourt,~FranceG6"@%4-%%typeG6$9$%%listG@%4-F46$9%F7-% &evalrG6#-%'SignumG6#,&F6"""F8)-F>6#-FA6#&F<6#""$@)/FHFDC$>FH-F>6#-FA6#F 6@%0FH""!FH,$-F"6$&F<6#""#FZFE/FHFEFen/FHFZ-F"6$F6Fhn-%&ERRORG6#%%FAILG@%F9C$>F H-F>6#-FA6#&F6FN@)FQC$>FH-F>6#-FA6#F<@%FY,$FHFE-F"6$&F6FinFZF[oFdpF\o-F"6$FfpF< F_oC'@'2&F66#FD&F8%F6>8&7%F\qFFaq7%F^qF6FZ>FcqFFaqF6>Fc qF<>8'-%%nopsG6#Faq>8(-F`r6#Fcq?(8$FOFjn-%$minG6$F^rFcr%%trueGC$>FH-F>6#-FA6#,& &Faq6#FgrFD&FcqFdsFE@)F[o-%'RETURNG6#-F"6$&Faq6#,&FgrFDFEFDFZFQ-Fhs6#-F"6$FZ&Fc qF]tF\oC$>FH-F"6$F\tFct@$FY-Fhs6#FHF_o@'/F^rFcr-Fhs6#FZ2F^rFcr-F"6$FZ&Fcq6#,&F^ rFDFDFD-F"6$&Faq6#,&FcrFDFDFDFZF0F0F0F0 M6R0 I(dev/devR6%%$fctG%"pG%"nG6(%&interG%%deveG%(fctnameG%"iG%"jG%"fG6$%)rememberG% RCopyright~Bruno~Salvy,~INRIA~Rocquencourt,~FranceG6"@0-%%typeG6$9$%%listG-%&ER RORG6#F74-%$hasG6$F7<$%"OG%&_XasyG@%/-%)zerotestGF;%%trueG""!F73-F56$F7-%(ratpo lyG6$%)anythingGFB4-F>6$F7FA-%0dev/dev/ratpolyG6$F79&-F56$F7%"+G-%*dev/dev/+G6% F79%FV-F56$F7%"*G-%*dev/dev/*GFfn-F56$F7%"^G@%-F>6$-%#opG6$""#F7FB-F"6%-%$expG6 #*&Fco"""-%#lnG6#-Fdo6$F]pF7F]pFgnFV-%,dev/dev/powG6&FapFcoFgnFV-F56$F7%)functi onGC$>8&-Fdo6$FHF7@+/F[q.F_p-%+dev/dev/lnG6%FapFgnFV/F[q.Fjo@%-F56$-FdoF;-%#&*G 6#<$-F`q6#FO-%'freeofG6#FBC$>8$-%'indetsG6$Fiq-F_p6#-%*dependentGFbr@%/-%%nopsG 6#FerF]pC$>Fer-FdoFas-F"6%)Fer*&FiqF]pFer!""FgnFV-%,dev/dev/expG6%FiqFgnFVFjs/F [qFAC$>Fer-F"6%FiqF]pF]p@%/Fer%%FAILGFdt-%&dev/OGFas/F[q.%'RootOfGC&>8%-%+dev/R ootOfG6$FiqFgn@$/-F`s6#F\uF]p-%'RETURNG6#&F\u6#F]p?&8'F\uFG@$0-%'removeG6%F5<#- %$seqG6$&Fju6#,&8(FfoF]pF]p/Fgv;F]p-%%iquoG6$-F`s6#FjuFfo%*nonnegintG<"-FeuF^wF guC&>8)-%$catG6$%%dev/GF[q@$4-F56$Fdw%*procedureG-F:6$%3Not~implemented~:~GFdw> Fer-%$mapG6&F"7#FiqFgnFV@%-F>6$FerFdtFdtC$>F\u-Fdw6$FdsFV@'55554-F56$F\uF8/&F\u 6#Fbu%)infinityG2Fgn-%+dev/lengthGFcu1,&FgnF]p""$F]pFV-%(hastypeGF6F\u2FVF]z-F" 6%F7Fgn,&FVF]p-%$minG6$F^z,(FVF]pF]pF]pFjyFisF]p-%1dev/dev/simplifyG6&F7F\uFgnF VF2F2F2F2, I0dev/dev/ratpolyR6$F%F'6'F)F,%"tG%#niG%"oGF2F2C%>%)TestzeroGFF?(Fgv,&FgnF]p""% F]pF]pF2FGC$>Fer-%*traperrorG6#-%'seriesG6%-%%subsG6$/FB*$F[qFisF7F[qFgv@$30Fer %*lasterrorG5554-F56$FerF_\l0-Fdo6$,&F_sF]pFisF]pFer-FAFhu2,&FgnFfo""&F]pF_s-%' testeqG6#,&Fa\lF]p-%(convertG6$Fer%(polynomGFis[@%F^]lC%?(F2F]pF]pF2-FF6#-Fdo6$ F]pFer>Fer-%'subsopG6$/F]pFHFer>FjuF_s@%/-Fdo6$,&FjuF]pFisF]pFerFd]l7&F]p-Fbv6$ -Fdo6$F\uFer/F\u;F]p,&FjuF]p!"#F]pF]p-Fdo6$FjuFer7&F]p-Fbv6$Ff_l/F\u;F]pFjuFHFh yFerF2F2F2F2, I1dev/dev/simplifyR6&F%%$devGF&F'6#%'newfctG6#F1F2C$>Fer-F\\l6#-%)simplifyG6#-% 'normalG6#-%%evalG6#-Fb\l6%/%"LGR6#%"xGF2F2F2-%$logG6#*$,&F]pF]pF7FisFisF2F2F2/ %"QGRF\blF2F2F2FablF2F2F2F7@%33Fh\l0FerF7/-F\\l6#-FF6$FerF7FG-F"6%FerFV9'FgnF2F 2F2F2, I*dev/dev/+RF$6%F*F,F)Fi`lF2C%>FerFH?&F\uF7FGC$>F[q-F"6%F\uFgnFV@%5/F[qFdt-F56$ F[q%*undefinedGC$>FerF[qF`^l>Fer-%(dev/addG6$FerF[q@'5554-F56$FerF8/&Fer6#F_sFh y2Fgn-F[zFas-%'memberG6$FG-Fcx6%F5FerF8FerFaz-F"6%F7Fgn,&FVF]p-Ffz6$F^z,(FVF]pF ]pF]pFcelFisF]p-Fjz6&F7FerFgnFVF2F2F2F2, I*dev/dev/*RF$FdclFi`lF2C%>FerF]p?&F\uF7FGC$>F[qFjcl@%F]dlC$>FerF[qF`^l>Fer-%(d ev/prdGFgdl@'FidlFerFazFielF_flF2F2F2F2, I,dev/dev/powR6&%"AG%"BGF&F'6%%$sigGF*F)Fi`lF2C$>Fer-%&evalrG6#-%'SignumG6#Fgn@ %3/FerFis-F56$F7.-FjoF_rC$>F\u-F\\l6#-F"6%F7FVFacl@%30F\uFi\l0F\uFdt-%(dev/powG 6%F\uFgnFacl-F"6%)-Fjo6#,$FiqFis,$FgnFis,&FVF]pF]pF]p,&FaclF]pF]pF]pC$>F[qFihl@ %F]dlF[qC$>F\u-F`il6%F[qFgnFacl@)5555/F\uFdtFbyFey2FVFjy-F`z6$)F7FgnF8F\u2Facl, &FVF]pFfoF]p-F"6%F[[mFV,&FaclF]p-Ffz6$F^z,(F]pF]pFaclF]pFjyFisF]p4Fijl-Fjz6&F[[ mF\uFVFaclF\uF2F2F2F2, I+dev/dev/lnRF$6$F*F)Fi`lF2C$>F\u-F"6%F7FgnFiil@%Fgjl-Fjz6&-F`qF;FdtFgnFVC$>Fer -%'dev/lnG6$F\uFV@%5FjdlF\zFer-F"6%-F_pF;FgnF[flF2F2F2F2, I,dev/dev/expRF$6$F)F*Fi`lF2C$>Fer-F"Ffn@%5Fct-F56$FerFadlFerC$>F\u-F\\l6#-%(de v/expG6$FerFV@'0F\u%;Not~a~convergent~expansionG@)/F\uFi\l-F:FcuF^yF\uFaz-F"6%- FjoF;FgnFdz-Fjz6&Fe^mF\uFgnFVFaz-F"6%Fe^mFgnF]zFdtF2F2F2F2, I(_equivX=F2%&falseGE\[l"F]p*$FBFisF2 M6R0 I-dev/endofdevR6%%$devG%"kG%$PolG67%"nG%$resG%'posminG%%miniG%"jG%$indG%$posG%$ valG%+candidatesG%'resultG%'intermG%"iG%(comblinG%"pG%)devcoeffG%&exactG%(currd egG%'degpolG%$polG%'tocompG%$sigG6#%RCopyright~Bruno~Salvy,~INRIA~Rocquencourt, ~FranceG6"@'/9$7'"""FEFE""!%)infinityG7$FE-%#opG6$;FE-%$minG6$,&9%""#FRFE-%%nop sG6#9&FV-%%typeG6$FC%*undefinedGFZC/>8$-FT6#FC@%/&FC6#FgnFGC$>83%%trueG>Fgn,&-% %iquoG6$FgnFRFE!""FEC$>F`o%&falseG>FgnFdo>8%7#<#7#-%"$G6$FFFgn>8+7#FF>&826#FapF E>8*FE@%52F\qFgnF`o@%-FX6$&FC6#,$F\qFR%%listG>&Fip6%-Fbp6$FF,&F\qFEFgoFEFE-Fbp6 $FF,&FgnFEF\qFgo7%-FJ6#FcqFFFG>FhqFcq@%Faq>FhqFcq>FhqFE>&8)6#FEFE@%/&FV6#FSFGC% >85&FV6#,&FSFE!"#FE>86-%&arrayG6$;FFFas%'sparseG?(8/FEFR,&FSFE!"$FEFao>&Fgs6#&F V6#,&F^tFEFEFE&FV6#F^tC%>FasF]s>FgsFhs?(F^tFEFR,&FSFEFgoFEFao>FbtFgt@%0&Fgs6#FF FF>8-7%&FCFjrFauFF>Fdu7#Ffu>84FF?(F@FEFEF@332-%+dev/lengthG6#Fdu,&FQFEFEFE2FjuF as55F`oF_q/&FirF]oFEC->8',&&FC6#""$FE&Fep6#FhrFE>8&7#FE?(8(FRFE-FN6$F\qFgnFaoC% >87,&&FC6#,&FewFRFEFEFE&Fep6#&Fir6#FewFE>88-%&evalrG6#-%'SignumG6#,&FjwFEFjvFgo @&/FdxFgoC$>FjvFjw>Fbw7#Few/FdxFF>Fbw7$-FJ6#FbwFew@$3/&Fbw6#-FTFfyF\qF_qC%>F\q, &F\qFEFEFE>&Fir6#F\qFE@%F^q@%Faq>FhqF`r>FhqFcq@%Faq>FhqFcq>FhqFE>8,<">8.FF?(Few FEFEF\zFaoC$?&F^t&F]p6#&Fir6#&FbwFbxFaoC$>807%-FJ6$;FE,&Ff[lFEFgoFEF^t,&&F^tFe[ lFEFEFE-FJ6$;,&Ff[lFEFEFEFgnF^t@$4-%'memberG6$Fi[lF[[lC&>F[[l-%&unionG6$F[[l<#F i[l@$0F^tF`p>&Fip6#-FJ6#Fi[l-%(dev/prdG6$&Fip6#-FJFht&Fip6%-Fbp6$FFF^\lFE-Fbp6$ FF,&FgnFEFf[lFgo>Fju-%(convertG6$7#-%$seqG6$&Fi[l6#81/F^_l;FEFgn%"+G@$31FjuFas0 &Fgs6#FjuFF>F^[l-%(dev/addG6$F^[l-%/dev/multbyrealG6$Fc]l*(Ff_lFE-%*factorialGF g_lFE-Ff^l6$-%$mapG6$Fa`lFi[l%"*GFgo>Fd[l,&Fd[lFEFEFE>F]p7$-FJ6#F]pF[[l>Fep7$-F J6#FepFjv@%5F^qFfv@$-FX6$F^[lFfq>F^[l7#-FJ6$;FE,&-FT6#F^[lFEFesFEF^[l@$4Feal>F^ [lFE@$0F^[lFF>Fdu7%-FJFavF^[lFjv@%33F^vF`o/F\qFgn7%FfblFFFGFduF@F@F@F@ M6R0 I.dev/factorialR6$%$devG%"nG6"6#%RCopyright~Bruno~Salvy,~INRIA~Rocquencourt,~Fr anceGF'-%*dev/GAMMAG6$-%(dev/addG6$9$"""9%F'F'F'F' M6R0 I-dev/implicitR6%%&expr1G%&expr2G%"pG6&%"xG%"kG%#d1G%#d2G6#%RCopyright~Bruno~Sa lvy,~INRIA~Rocquencourt,~FranceG6"C$@$5-%%typeG6$9$%*undefinedG-F46$9%F7-%'RETU RNG6#F7@%/&F66#""$""!@)3/&F:FAFC/&F66#""#&F:FJ-F"6%-%'subsopG6%/FK%%NULLG/FBFSF 6-FP6%FRFTF:9&30FGFC-F46$FI%%listG-F"6%7%-%#opG6#FIFC%)infinityGF:FW-F46$7$-F[o 6$FBF:-F[o6$FKF:7$FCFfn-F"6%F67%-F[o6#FLFCF]oFW-%&ERRORG6#%2no~solution~foundG@ %0F@!""@%-F46$F@%'negintGC%>8&-%(dev/powG6%F6,$*$F@FapFapFW>8'-Fjp6%F:F\qFW7#-% $seqG6$-F[o6#-F"6%Fhp-%/dev/multbyrealG6$F_q-%$expG6#**^#!"#"""8%Fcr%#PiGFcrF@F apFW/Fdr;FC,&F@FapFapFcr-F"6%FipF`qFW@)FZ-F"6%-%'dev/lnG6$F6FW-F_s6$F:FWFW0-%)z erotestG6$FIFcr%%trueG-F"6%-F[r6$F6*$FIFap-F[r6$F:F\tFW32Fcr-%+dev/lengthG6#F60 &F66#""&F]o@%0&F66#FcrFcr-F"6%7'FjtFcrFapFCF]o-%5dev/implicit/itersumG6%-%(dev/ devG6%-%%subsG6$/%&_XasyG-F[o6#-F"6%F^u7'FcrFcrFapFCF]oFW-%%evalG6#-Ffu6$/%)_Xa sytabG%(_equivXG-%,dev/instancG6#-%(dev/addG6$7%-F[o6$;FcrFBF6FCF]o-F[r6$F6FapF WFWF:FWFW7#-F`u6%-F[r6$FOFapF:FW@%/FjtFcr-F<6#7#F:C$>Fhp-Ffu6$/Fiu8$-F`v6#-Ffu6 $Fdv*$-Fhv6$7%FjtFcrFcrFbxFap@'/-F[o6$FCFhpF^r-F"6%-F_s6$F^uFWFasFW/F]y%#lnG-F" 6%-%(dev/expGFby-FhyFbsFW-F\p6#%2should~not~happenGF/F/F/F/, I5dev/implicit/itersumR6%%*othersideG%$valGF'6$%%exprG%$resGF-F/C%>Fbx-F`v6#-Ff u6$Fdv-FhvFct>FdrF:?(F/FcrFcrFWFgs>Fdr-%+dev/reduceG6$-F[w6$F:-Fcu6%-Ffu6$/Fiu- %*dev/printG6%FdrFiuF]oFbxFWFWFWF/F/F/F/ M6R0 I-dev/indexifyR6#%%exprG6'%"iG%$cstG%$fcnG%&toindG%$resG6#%RCopyright~Bruno~Sal vy,~INRIA~Rocquencourt,~FranceG6"@%/-%#opG6$""!9$%#lnGC$>8(7#"""?(8$""#F;F.0&F9 6#-%%nopsG6#F9%)infinityG>F9-%(dev/devG6%-%)simplifyG6%-F66#&%(_equivXG6#-F26#- F26#F5F6.%)symbolicGF=F=@'-%%typeG6$F5%%listG-%'subsopG6$/""$!""-%,dev/placeitG 6#-%%evalG6#-%%subsG6$/%)_XasytabGFP-%,dev/instancGFU-FZ6$F5-.%$expG6#FfnC$>8'- Fao6#-Fdo6$Ffo-FioFS@%-FZ6$Fbp%"*GC&>8&F;>8%F;?&F=Fbp%%trueG@%-%$hasG6$F=%&_Xas yG>F^q*&F^qF;F=F;>F`q*&F`qF;F=F;-Fhn6$/F[o,$F`qF\o-F^o6#-F^p6#F^q-Fhn6$Fjn-F^o6 #-F^p6#Fbp-%&ERRORG6#.%$bugGF.F.F.F., I,dev/placeitRF$6#%$ordG6$%)rememberGF-F.C$>F=-%+dev/LorderGFU-%.dev/findplaceG 6%F5,$F=F\oF=F.F.F.F., I+dev/LorderR6#%$devGF.F,F.@+45-Feq6$F5F6-Feq6$F5F^pF4-FZ6$F5%)functionG,&F;F;- 9!FSF;-FZ6$F5<$F[q%"+G-%$maxG6$-F]u6#-F26$F;F5-F]u6#-F26$F>F5-FZ6$F5%"^G,&FeuF; FiuF;-F[s6$%2invalid~expansionGF5F.F.F.F., I.dev/findplaceR6%%%devtG%%lim1G%%lim2G6$F'%$sigGF,F.@+3-FZ6$7$9%9&7$%(integerG Fdw2F;,&FbwF;FawF\oC&>F=-%&truncG6#,&Faw#F;F>FbwF]x@$3-FZ6$F=Fdw4-%)assignedG6# &FP6#F=@%1F=F4C%>Ffx-F^p6#Fgq?(F.F;F;F\tFbq>Ffx-F^pFex>Ffx*$FfxF\oC%>Ffx-F6F]y? (F.F;F;,&F=F;!"#F;Fbq>Ffx-F6Fex>FfxFby>F`q-%1dev/comparexplogG6$F5-Fao6#-Fdo6$F fo-Fio6#7%F=F;F\o@'/F`qF;-F]u6%F5FawF=/F`qF47'F=F;F\oF4FE-F]u6%F5F=Fbw/F5*$-Fao 6#-Fdo6$Ffo-Fio6#7%FawF;F;F\o7'FawF;F\oF4FE/F5*$-Fao6#-Fdo6$Ffo-Fio6#7%FbwF;F;F \o7'FbwF;F\oF4FE-Fdx6#&FPF[xC$>F`q-F^z6$F5*$-Fao6#-Fdo6$Ffo-Fio6#7%F\xF;F;F\o@' F[[l7'F\xF;F\oF4FEFhz-F]u6%F5FawF\x-F]u6%F5F\xFbwC$>Fe\l*$F5F\oFc]lF.6#FPF.F., I1dev/comparexplogR6$%"uG%"vG6#F[wF,F.@//F5FawF445-Feq6$F5Fgq-Feq6$FawFgqC$>F=- %&evalrG6#-%'SignumG6#,&F5F;FawF\o@%/F=%%FAILG-F[s6#Fe_lF=3-FZ6$7$F5Faw7$FjtFjt /F1-F26$F4Faw-F]u6$Fgu-F26$F;Faw-FZ6$F5.-F^p6#%)anythingG-F]u6$FT-%'expandG6#-F 66#Faw3-FZ6$F5.-F6Fh`l0-F]alFUF5-%,dev/compareG6$-FH6%FgalF;F;-FH6%FawF;F;-FZ6$ F5<&F[qF_vFau%%nameG-Fial6$-FH6%-F]al6#-F6FUF;F;-FH6%F\alF;F;FavF.F.F.F., I.dev/exp/truncR6#%"xG6#F+F,F.@%-FZ6$F5Fdw@%2F4F5F5F;C$>F=,&-Fjw6#-%&evalfGFUF; F;F;@%2F4F=F=F;F.F.F.F. M6R0 I+dev/lcoeffR6#%$devG6#%"uG6#%RCopyright~Bruno~Salvy,~INRIA~Rocquencourt,~Franc eG6"@'4-%%typeG6$9$%%listGF0/-%%nopsG6#F0""$@%4-F.6$-%#opG6$""#F0F1%%FAILG-F"6# F;C%>8$F0?(F*"""FFF*-F.6$FDF1>FD-F<6$F>FDFDF*F*F*F* M6R0 I*dev/printR6%%$devG%$varG%&globnG6&%$resG%%bigOG%#nbG%"nG6#%RCopyright~Bruno~S alvy,~INRIA~Rocquencourt,~FranceG6"@%-%%typeG6$9$%*undefinedGF5C%@%-F26$F4%%lis tGC$>8'-%$minG6$9&,&-%+dev/lengthG6#F4"""!""FF>8$-%1dev/print/nfirstG6&F4,&F=FF FFFF.8&.8%>FI-FK6&F4""!FN.%%big0G@$3-%)assignedG6#FQ0FQFU>FI7$-%#opG6#FI-%+simp lify/OG6$-%"OG6#-%%evalG6#-%%subsG6$/%)_XasytabG%(_equivXG-%)simplifyG6$FQ%)inf inityG%&_XasyG-%(convertG6$-Fgo6$/F`p9%FI%"+GF/F/F/F/, I1dev/print/nfirstR6&%%treeGF,%&totalGF*6*F&%"iGF+%%locOG%"kGF)%%restG%&interGF -F/@%4F8@%/F4FUC%>9'FU>FAFU7"C%>FAFF@$/FgpFF>FjqFF-Fdo6#-Fgo6$Fio7#F4C(>FI-Fdo6 #-Fgo6$Fio&Fjo6#&F46#FF>8*Fgp>8)F\r?(FQ""#Ffs-%%nopsGFE2FUFbsC$>8+-FK6&&F4FfnFb sFN.F=@%4-Fen6#F=C$>Fds7$-F[o6#Fds-%$seqG6$*&&F\t6#8(FF-%)frontendG6%F]p7%)FI&F 46#,&FQFFFFFF%&powerG%$expG7$<%%"*G%"^GFhp<$%#lnGFjuFF/F`u;FFFO>Fbs,&FbsFFFOFGC %>Fds7$Fht-F[u6$F]u/F`u;FF,&FOFFFGFF>FbsFU>Fjq*&F=FFFauFF>FA,&FgpFFFbsFGFdsF/F/ F/F/, I+simplify/OR6$%"fGF&6#%"aGF-F/@)54-%$hasG6$F4Fao/9#FFF4-F26$F4.-Fao6#%)anythin gGC%>FI-F[oFE?(F/FFFFFfs%%trueG>FI-%/simplify/O/inOG6$FIFgp@%0FIFU-FaoF\oFU-F26 $F4F]v-F^o6$-Fao6#-Fbp6$-%$mapG6$F[o-%'selectG6%F27#FhxFbxF]vFgp-F[z6%F^oF4FgpF /F/F/F/, I/simplify/O/inOR6$%%exprGF&6#FewF-F/@33-F26$F4%)constantG4-F\x6$F4F_p@%-%'test eqGFEFUFF-F26$F4%%nameG@%/F4FgpFgpFF-F26$F4FhpC$>FI-F[z6%9!F4Fgp@%0FIF4-F^\lF^y F4-F26$F4F^vC$>FI-%&evalcG6#-F[o6$FfsF4@%-F\x6$FI^#FF-F^\l6$)-F[o6$FFF4-%&coeff G6%FIF^]lFUFgp)Fb]lFIFbyF\\l-F26$F4.-FjuFdxC$>FI-Fg\l6#Fhx@%F\]l-F^\l6$-Fju6#Fd ]lFgpF4-F26$F4<$.-%$sinGFdx.-%$cosGFdxFF-F26$F4%&rangeG-F^\l6$Fi\lFgpC$>FI-Fg\l FE@%F`\lFa\lF4F/F/F/F/ M6R0 I-dev/realpartR6#%%exprG6&%"iG%'resultG%"nG%$auxG6#%RCopyright~Bruno~Salvy,~INR IA~Rocquencourt,~FranceG6"@%4-%%typeG6$9$%%listG@%-F16$F3%)realconsGF3-%&evalcG 6#-%#ReG6#F3C&>8&-%%nopsGF>?(8$""#FF,&FA"""!"$FH%%trueGC$@%-F16$&F36#FEF4C$>8'- F"6#7%-%#opG6#FO""!%)infinityG@$3-F16$FSF4/&FS6#-FC6#FSFen>FS-%'subsopG6%/F]o%% NULLG/,&F]oFH!""FHFdoFS>FS-F"FY@%0FSFZ>&8%FP6$FS&F36#,&FEFHFHFH>F]pFdo>F^p-%$se qG6$&F^p6#,$FEFF/FE;FH-%%iquoG6$FGFF@'/&F36#FAFen@'/F^pFdoFZ3/-FC6#7#F^pFF/&F^p 6#FFFZ&F^p6#FH7&&F3F_rF^pFZFen-F16$&F36#,&FAFHFgoFHF47&FarF^p-F"6#FdrFbq7&FarF^ pFHFbqF-F-F-F- M6R0 I'dev/SiR6$%"uG%"nG6/%%factG%"iG%"xG%(exampleG%&sinusG%(cosinusG%"jG%"kG%%initG %$sigG%%invuG%$resG%%termG6#%RCopyright~Bruno~Salvy,~INRIA~Rocquencourt,~France G6"@'-%%typeG6$9$%*undefinedGF=4-F:6$F<%%listGC$>8,-%*traperrorG6#-%#SiG6#F<@%/ FD%*lasterrorGF=FDC$>8--%&evalrG6#-%'SignumG6#&F<6#""$@)/FP"""C%>8*#!""""'?(8%" "#Ffn9%%%trueGC$>Fin,$*(FinFfn,&F^o""%F_oFfnF[oF^oF[oF[o>&8'6#F^o6$*&FinFfn,&F^ oF_oFfnFfnF[oF^o-%(dev/prdG6$F<-%-dev/endofdevG6%-%(dev/powG6%FFP-FR6#-FU6#-%#ReG6#-%+dev/lcoeffGF J@'5Faq/FPFipF=/FP%%FAILG-%&ERRORG6#FarC*>8$-Ffp6%F8)-%+dev/cossinG6$F< F`o>8.-Ffp6%F8(-F`p6$&F\s6#F_oFas>F\s&F\s6#Ffn>8/-%(dev/addG6$,$%#PiG#F fnF_o-F`t6$-F`p6$-%/dev/multbyrealG6$F\sF[oFas-F`p6$-Fjt6$FesF[oFas?(F^oFfnFfn, &-%%iquoG6$,&F`oFfnFfnFfnF_oFfnF[oFfnFaoC$>80-F`p6$FhuFgr>F^t-F`t6$F^t-F`t6$-F` p6$-Fjt6$F\s*&)F[o,&F^oFfnFfnFfnFfn-%*factorialG6#,$F^oF_oFfnFhu-F`p6$-Fjt6$Fes *&FevFfn-Fhv6#F^pFfnFhu-%+dev/reduceG6$F^tF`oF_rC$@%-F:6$&FFD-F"6$7%-%#o pG6#FiwFip%)infinityGF`o>FD-F"6$FiwF`o@%32F`o-%+dev/lengthG6#FD0&FD6#-%%nopsGFj xFaxFDC&>Fin-FI6#8&>8+Ffn?(F^oFfnFfnF`oFaoC%>Fin-%%diffG6$FinFdy>Ffy*&FfyFfnF^o F[o>Fio6$*&-%%subsG6$/FdyFiwFinFfnFfyFfnF^o-Fcp6%-%'subsopG6%/F_o%%NULLG/FYF\[l F8,-%*traperrorG6#-%#CiG6#F?@%/FK%*lasterrorGFA-FB6#FK>8--%&evalrG6#-%'Signu mG6#&F?6#""$@)/FX"""C(>8$-%(dev/powG6%F?""#9%>8/F^o>8.""!>80-%(dev/addG6$%&gamm aG-%'dev/lnG6$F?Ffo?(8%F^oF^o,&FfoF^o!""F^o0FjoF]pC%>FjoF]p>Fho-%(dev/prdG6$Fho Fao>F]p-F_p6$F]p-%/dev/multbyrealG6$Fho,$*()FhpFfpF^o-%*factorialG6#,$FfpFeoFhp FfpFhp#F^oFeo-FB6#-%+dev/reduceG6$F]pFfo/FXFhpC->FX-FZ6#-Fgn6#-%#ReG6#-%+dev/lc oeffGFQ@&5Fcr/FXF[pFA/FX%%FAILG-%&ERRORG6#Fcs>81-Fco6%F?FhpFfo>Fao-Fco6%FhsFeoF fo>8(-%+dev/cossinGFdp>8)-F^q6$&F_t6#F^oFhs>F_t&F_t6#Feo>Fjo-F_p6$-F^q6$F_tFao- F^q6$Fao-Fdq6$FctFhp>82F^o?(FfpF^oF^o-%&truncG6#,&FfoF]rF]rF^o%%trueGC$>Feu-F^q 6$FeuFao>Fjo-F_p6$Fjo-F_p6$-F^q6$-Fdq6$F_t*&FhqF^oFiqF^oFeu-F^q6$-Fdq6$Fct*&)Fh p,&FfpF^oF^oF^oF^o-Fjq6#,&FfpFeoF^oF^oF^oFeu-FB6#-Far6$FjoFfoFasC$@%-F=6$&F?Fjt FH>FK-F"6$7%-%#opG6#F\xF[p%)infinityGFfo>FK-F"6$F\xFfo@%32Ffo-%+dev/lengthGFV0& FK6#-%%nopsGFVFdxFUC'>8'7$FKF[p>8*-FP6#8&>8+F^o?(FfpF^oF^oFfoF[vC%>Fgy-%%diffG6 $FgyFjy>F\z*&F\zF^oFfpFhp>Fdy7%-Fbx6#Fdy*&-%%subsG6$/FjyF\xFgyF^oF\zF^oFfp-FB6# -%-dev/endofdevG6%FaoFfoFdyFdsF9F9F9F9 M6R0 I'dev/EiR6$%"uG%"nG6-%%factG%"iG%"xG%"jG%(exampleG%"kG%%initG%$sigG%'tomultG%$r esG%'newresG6#%RCopyright~Bruno~Salvy,~INRIA~Rocquencourt,~FranceG6"C&@$-%%type G6$9$%*undefinedG-%'RETURNG6#F<@$4-F96$F;%%listGC$>8*-%*traperrorG6#-%#EiG6#F;@ %/FG%*lasterrorGF=-F>6#FG>8+-%&evalrG6#-%'SignumG6#&F;6#""$@)/FT"""C*>FT-FV6#-F Y6#-%#ReG6#-%+dev/lcoeffGFM@(/FT!""-F>6#-%/dev/multbyrealG6$-F"6$-F\p6$F;Fho9%F ho/FT""!F=/FT%%FAILG-%&ERRORG6#Ffp>8$F;>8,Fjn>8-Fdp>8.-%(dev/addG6$-%'dev/lnG6$ F;Fbp%&gammaG?(8%FjnFjn,&FbpFjnFhoFjn0F_qFaqC%>F_qFaq>F]q-%(dev/prdG6$F]qF[q>Fa q-Fcq6$Faq-F\p6$F]q*&FjqFho-%*factorialG6#FjqFho-F>6#-%+dev/reduceG6$FaqFbpFgoC '>F[q-%(dev/powG6%F;FhoFbp>8(7.FjnFjnFjn""#FisFgn""'""%"#C""&"$?"Fjs>8'F^t?(Fjq FjsFjnFbp%%trueGC$>F`t*&F`tFjnFjqFjn>Fgs7%-%#opG6#FgsF`t,&FjqFjnFjnFjn-F>6#-Far 6$-%(dev/expGFgq-%-dev/endofdevG6%F[qFbpFgsFcpC$@%-F96$&F;6#FisFD>FG-F"6$7%-Fit 6#FiuFdp%)infinityGFbp>FG-F"6$FiuFbp@%32Fbp-%+dev/lengthGFR0&FG6#-%%nopsGFRFavF QC(>Fgs7$FGFdp>F`t-FL6#8&>8)Fjn?(FjqFjnFjnFbpFbtC%>F`t-%%diffG6$F`tFew>Fgw*&Fgw FjnFjqFho>Fgs7%Fht*&-%%subsG6$/FewFiuF`tFjnFgwFjnFjq>F[q-%'subsopG6%/Fis%%NULLG /FgnF\yF;-F>6#FbuFgpF5F5F5F5 M6R0 I-dev/FresnelSR6$%"uG%"nG60%%factG%"iG%"xG%(exampleG%(cosinusG%&sinusG%"kG%"jG% %initG%$sigG%%invuG%'newresG%$resG%'tomultG6#%RCopyright~Bruno~Salvy,~INRIA~Roc quencourt,~FranceG6"C&@$-%%typeG6$9$%*undefinedG-%'RETURNG6#F?@$4-F<6$F>%%listG C$>8,-%*traperrorG6#-%)FresnelSG6#F>@%/FJ%*lasterrorGF@-FA6#FJ>8--%&evalrG6#-%' SignumG6#&F>6#""$@)/FW"""C&>8'7$,$%#PiG#F]o""'""!>8*,$Fco#F]o""#?(8%F]oF]o9%%%t rueGC$>Fho,$**FhoF]oFcoF[pF]p!"",&F]pF[pF]oF]oFdp#Fdp"")>F`o7%-%#opG6#F`o*&FhoF ]o,&F]p""%FjnF]oFdpF]p-FA6#-%(dev/prdG6$-%(dev/powG6%F>FjnF^p-%-dev/endofdevG6% -Ffq6%F>F_qF^pF^pF`o/FWFdpC/>FW-FY6#-Ffn6#-%#ReG6#-%+dev/lcoeffGFP@&5F]r/FWFfoF @/FW%%FAILG-%&ERRORG6#F]s>8)-%+dev/cossinG6$-%/dev/multbyrealG6$-Ffq6%F>F[pF^pF ioF^p>8(&Fbs6#F]o>8.-Ffq6%F>FdpF^p>8$-Ffq6%F`tF[pF^p>Fbs-Fcq6$&Fbs6#F[pFdt>Fdt- Ffq6%FdtF[pF^p>81F]o>80Ffo>8/-%(dev/addG6$Fjo-Ffu6$-Fgs6$-Fcq6$F\tF`t,$*$FcoFdp Fdp-Fgs6$-Fcq6$FbsF`t,$*$Fco!"#Fdp?(F]pF]oF]o-%&truncG6#,&F^pFjoFjoF]o0FbuFduC% >FbuFdu>F`u-Fcq6$F`uFdt>Fdu-Ffu6$Fbu-Ffu6$-Fgs6$-Fcq6$F\tF`u*()Fdp,&F]pF]oF]oF] oF]o-%(productG6$,&.FhoF[pFdpF]o/Fbx;F]o,$F]pF[pF]o)FcoFepFdp-Fgs6$-Fcq6$FbsF`u *(F\xF]o-F_x6$Fax/Fbx;F]oFepF]o)Fco,&F]pF[pF[pF]oFdp-FA6#-%+dev/reduceG6$FduF^p F[sC$@%-F<6$&F>F[uFG>FJ-F"6$7%-F[q6#F[zFfo%)infinityGF^p>FJ-F"6$F[zF^p@%32F^p-% +dev/lengthGFU0&FJ6#-%%nopsGFUFbzFTC(>F`o7$FJFfo>8+-FO6#8&>FhoF]o?(F]pF]oF]oF^p F_pC%>Fd[l-%%diffG6$Fd[lFg[l>Fho*&FhoF]oF]pFdp>F`o7%Fjp*&-%%subsG6$/Fg[lF[zFd[l F]oFhoF]oF]p>Fdt-%'subsopG6%/F[p%%NULLG/FjnF]]lF>-FA6#-Fiq6%FdtF^pF`oF^sF8F8F8F 8 M6R0 I-dev/FresnelCR6$%"uG%"nG60%%factG%"iG%"xG%(exampleG%&sinusG%(cosinusG%"kG%"jG% %initG%%invuG%'newresG%$resG%$sigG%'tomultG6#%RCopyright~Bruno~Salvy,~INRIA~Roc quencourt,~FranceG6"C&@$-%%typeG6$9$%*undefinedG-%'RETURNG6#F?@$4-F<6$F>%%listG C$>8,-%*traperrorG6#-%)FresnelCG6#F>@%/FJ%*lasterrorGF@-FA6#FJ>80-%&evalrG6#-%' SignumG6#&F>6#""$@)/FW"""C&>8'7&F]o""!,$*$%#PiG""##!"""#SF]o>8*,$Fdo#Fho"")?(8% FfoF]o9%%%trueGC$>F[p,$**F[pF]oFeoFfoF`pFho,&F`pFfoFhoF]oFhoF]p>F`o7%-%#opG6#F` o*&F[pF]o,&F`p""%F]oF]oFhoF`p-FA6#-%(dev/prdG6$F>-%-dev/endofdevG6%-%(dev/powG6 %F>F_qFapFapF`o/FWFhoC/>FW-FY6#-Ffn6#-%#ReG6#-%+dev/lcoeffGFP@&5F[r/FWFboF@/FW% %FAILG-%&ERRORG6#F[s>8)-%+dev/cossinG6$-%/dev/multbyrealG6$-Fiq6%F>FfoFap,$Feo# F]oFfoFap>8(&F`s6#Ffo>8--Fiq6%F>FhoFap>8$-Fiq6%F`tFfoFap>F`s-Fcq6$&F`s6#F]oFdt> Fdt-Fiq6%FdtFfoFap>8/Fbo>81F]o>8.-%(dev/addG6$Fjs-Ffu6$-Fes6$-Fcq6$F\tF`t*$FeoF ho-Fes6$-Fcq6$F`sF`t,$*$Feo!"#Fho?(F`pF]oF]o-%&truncG6#,&FapFjsFjsF]o0FduF`uC%> F`uFdu>Fbu-Fcq6$FbuFdt>Fdu-Ffu6$F`u-Ffu6$-Fes6$-Fcq6$F\tFbu*()FhoF`pF]o-%(produ ctG6$,&.F[pFfoFhoF]o/F`x;F]o,$F`pFfoF]o)Feo,&F`pFfoF]oF]oFho-Fes6$-Fcq6$F`sFbu* ()Fho,&F`pF]oF]oF]oF]o-F]x6$F_x/F`x;F]oFexF]o)Feo,&F`pFfoFfoF]oFho-FA6#-%+dev/r educeG6$FduFapFirC$@%-F<6$&F>F^tFG>FJ-F"6$7%-F[q6#F\zFbo%)infinityGFap>FJ-F"6$F \zFap@%32Fap-%+dev/lengthGFU0&FJ6#-%%nopsGFUFczFTC(>F`o7$FJFbo>8+-FO6#8&>F[pF]o ?(F`pF]oF]oFapFbpC%>Fe[l-%%diffG6$Fe[lFh[l>F[p*&F[pF]oF`pFho>F`o7%Fjp*&-%%subsG 6$/Fh[lF\zFe[lF]oF[pF]oF`p>Fdt-%'subsopG6%/Ffo%%NULLG/FjnF^]lF>-FA6#-Ffq6%FdtFa pF`oF\sF8F8F8F8 M6R0 I-dev/FresnelfR6$%"uG%"nG6*%%factG%"iG%"xG%(exampleG%"kG%"jG%%initG%$sigG6#%RCo pyright~Bruno~Salvy,~INRIA~Rocquencourt,~FranceG6"C&@$-%%typeG6$9$%*undefinedG- %'RETURNG6#F9@$4-F66$F8%%listGC$>8*-%*traperrorG6#-%)FresnelfG6#F8@%/FD%*laster rorGF:-F;6#FD>8+-%&evalrG6#-%'SignumG6#&F86#""$@)/FQ"""C&>8'7$%#PiG""!>8(F\o?(8 %FgnFgn,&9%Fgn!""Fgn%%trueGC$>F_o,$*(F_oFgnF\o""#,&Fao""%FgnFgnFdoFdo>Fjn7%-%#o pG6#FjnF_oFao-F;6#-%(dev/addG6$#FgnFjo-%(dev/prdG6$-%(dev/powG6%F8FZFbo-%-dev/e ndofdevG6%-F\q6%F8F\pFboFboFjn/FQFdoC(>FQ-FS6#-FV6#-%#ReG6#-%+dev/lcoeffGFJ@&5F cq/FQF]oF:/FQ%%FAILG-%&ERRORG6#Fcr>Fjn7&*$F\oFdoF]o,$*$F\o!"$F\sFgn>F_oFjr?(Fao FjoFgnFcoFeoC$>F_o,$**F_oFgn,&FaoF\pFdoFgnFgn,&FaoF\pF\sFgnFgnF\o!"#Fdo>FjnF^p- F;6#-Fip6$-F\q6%F8FdoFco-F_q6%-F\q6%F8!"%FcoFcoFjnFarC$@%-F66$&F86#FjoFA>FD-F"6 $7%-F`p6#FftF]o%)infinityGFco>FD-F"6$FftFco@%32Fco-%+dev/lengthGFO0&FD6#-%%nops GFOF^uFNC(>Fjn7$FDF]o>8)-%(FresnelG6#8&>F_oFgn?(FaoFgnFgnFcoFeoC%>F`v-%%diffG6$ F`vFdv>F_o*&F_oFgnFaoFdo>Fjn7%F_p*&-%%subsG6$/FdvFftF`vFgnF_oFgnFao>8$-%'subsop G6%/Fjo%%NULLG/FZF[xF8-F;6#-F_q6%FfwFcoFjnFdrF2F2F2F2 M6R0 I-dev/FresnelgR6$%"uG%"nG6*%%factG%"iG%"xG%(exampleG%"kG%"jG%%initG%$sigG6#%RCo pyright~Bruno~Salvy,~INRIA~Rocquencourt,~FranceG6"C&@$-%%typeG6$9$%*undefinedG- %'RETURNG6#F9@$4-F66$F8%%listGC$>8*-%*traperrorG6#-%)FresnelgG6#F8@%/FD%*laster rorGF:-F;6#FD>8+-%&evalrG6#-%'SignumG6#&F86#""$@)/FQ"""C&>8'7&!""""!,$*$%#PiG"" ##Fgn""&Fgn>8(F^o?(8%FaoFgn,&9%FgnF\oFgn%%trueGC$>Feo,$*(FeoFgnF`oFao,&Fgo""%Fg nFgnF\oF\o>Fjn7%-%#opG6#FjnFeoFgo-F;6#-%(dev/addG6$#FgnFao-%(dev/prdG6$F8-%-dev /endofdevG6%-%(dev/powG6%F8F`pFhoFhoFjn/FQF\oC(>FQ-FS6#-FV6#-%#ReG6#-%+dev/lcoe ffGFJ@&5Feq/FQF]oF:/FQ%%FAILG-%&ERRORG6#Fer>Fjn7&*$F`o!"#F]o,$*$F`o!"%!#:Fgn>Fe oF]s?(FgoFaoFgnFioFjoC$>Feo,$*(FeoFgn,&*$FgoFao"#;F\oFgnFgnF`oF\sF\o>FjnFbp-F;6 #-F]q6$-Fcq6%F8!"$Fio-F`q6%-Fcq6%F8F_sFioFioFjnFcrC$@%-F66$&F86#FaoFA>FD-F"6$7% -Fdp6#FjtF]o%)infinityGFio>FD-F"6$FjtFio@%32Fio-%+dev/lengthGFO0&FD6#-%%nopsGFO FbuFNC(>Fjn7$FDF]o>8)-FI6#8&>FeoFgn?(FgoFgnFgnFioFjoC%>Fdv-%%diffG6$FdvFgv>Feo* &FeoFgnFgoF\o>Fjn7%Fcp*&-%%subsG6$/FgvFjtFdvFgnFeoFgnFgo>8$-%'subsopG6%/Fao%%NU LLG/FZF^xF8-F;6#-F`q6%FiwFioFjnFfrF2F2F2F2 M6R0 I*dev/GAMMAR6$%"uG%"nG6.%%factG%"iG%"xG%(exampleG%%newuG%"jG%)example2G%*firstp artG%"kG%%initG%$sigG%#coG6$%)rememberG%RCopyright~Bruno~Salvy,~INRIA~Rocquenco urt,~FranceG6"C&@$-%%typeG6$9$%*undefinedG-%'RETURNG6#F>@$4-F;6$F=%%listGC$>8-- %*traperrorG6#-%&GAMMAG6#F=@%/FI%*lasterrorGF?-F@6#FI>8.-%&evalrG6#-%'SignumG6# &F=6#""$@)5/FV"""3/FV""!-F;6$&F=6#""#%'negintGC%@%F\oC%>8(F=>8%F`o>8'7'F]oF]o!" "F`o%)infinityGC&>F[p-%'subsopG6%/Feo%%NULLG/FinFipF=>F]pFco>F_p-%(dev/devG6%*$ -%'expandG6#-%(convertG6$7#-%$seqG6$,&F]oF]o*&8)F]o%&_XasyGF]oFap/F]r;F]o,$F]pF ap%"*GFap9%Fcr>F_p7$F]o-Fiq6$-%#opG6#7$&F_p6#,$F]rFeo,&F]rF]o!"#F]o/F]r;F]o-%%i quoG6$-%%nopsG6#F_pFeo>8*-%(dev/prdG6$F_p-F"6$7)F]oF]oF`oF]oF]oF`oFbpFcr-F@6#-% (dev/addG6$-%/dev/multbyrealG6$-%(dev/powG6%F[pFapFcr&FjsFdo-%-dev/endofdevG6%F [pFcr-Ffp6&/F]oFipFhpFjpFjs/FVFapC)>FV-FX6#-Fen6#-%#ReG6#-%+dev/lcoeffGFO@&FcuF ?/FV%%FAILG-%&ERRORG6#Fav>8+-Fgt6$-F\t6$-%(dev/expG6$-F\t6$F=-Fdt6$Fap-%'dev/ln G6$F=FcrFcr-Fjt6%F=#FapFeoFcr*&Feo#F]oFeo%#PiGFiw@$32Fcr-%+dev/lengthG6#Ffv0&Ff v6#-FgsF`xFbp-F@F`x>F_p-Ffp6$Fbu-F\w6$7$F]o-Fiq6$-Fir6#7$*(-%*bernoulliG6#,$F]p FeoF]oF]pFap,&F]p""%F`sF]oFap,&F]pFeoFapF]o/F]p;F]o,&FcrF]oF]oF]oF[z>8$-Fjt6%F= FapFcr-F@6#-F\t6$Ffv-F^u6%F]zFcrF_pF_o@'/FcoF]oC'>&F_p6#F`oF]o>&F_p6#F]oF`o>8/7 $,$%&gammaGFap-Fiq6$*&)Fap,&F]pF]oF]oF]oF]o-%%ZetaG6#Fh[lF]o/F]p;F]o,&FcrF]oFap F]o?(F]pF]oF]oFcr%%trueGC$>&F_pFdy*&-Feq6$7#-Fiq6$*&&F`[l6#,&F]rF]oF]oF]oF]o&F_ p6#,(F]pFeoF]rF`sF`sF]oF]o/F]r;F`o,&F]pF]oFapF]o%"+GF]oF]pFap>&F_p6#,&F]pFeoF]o F]oF]p-F@6#-F^u6%FepFcr7#-Fiq6$&F_p6#F]p/F]p;F`o,&FcrFeoF]oF]o3-F;6$Fco%(intege rG2F]oFcoC'>F_p-F^q6%-Fbq6#-Feq6$7#-Fiq6$,&F]oF]oF\rF]o/F]r;F]o,&FcoF]oFapF]oFb rFcrFcr>F_p7$F]o-Fiq6$-Fir6#7$F\s,&F]rF]oFapF]oFas@$2Fco,&FcrF]oFeoF]o>F_p-Ffp6 $/,&FcoFeoFinF]oFbpF_p>FjsF^t-F@6#-F^u6%FepFcr-Ffp6$Fbu-F\t6$F_pFjsC%@%-F;6$Fco FF>FI-F"6$7%-Fir6#FcoF`oFbpFcr>FI-F"6$FcoFcr@$-F;6$FIF>F?@%32Fcr-F_xFT0&FI6#-Fg sFTFbpFSC(>F_p7$FIF`o>F]r-.FN6#8&>8,F]o?(F]pF]oF]oFcrF`\lC%>F]r-%%diffG6$F]rFac l>Fccl*&FcclF]oF]pFap>F_p7%-FirFhs*&-%%subsG6$/FaclFcoF]rF]oFcclF]oF]p>F]zFep-F @6#FdzFbvF7F7F7F7 M6R0 I(dev/PsiR6$%"uG%"nG6.%%factG%"iG%"xG%(exampleG%%newuG%"jG%*firstpartG%)example 2G%"kG%%invuG%%initG%$sigG6#%RCopyright~Bruno~Salvy,~INRIA~Rocquencourt,~France G6"C'@$/9#""$-%'RETURNG6#-%.dev/polygammaG6#9"@$-%%typeG6$9$%*undefinedG-F=6#FH @$4-FE6$FG%%listGC$>8.-%*traperrorG6#-%$PsiG6#FG@%/FR%*lasterrorGFI-F=6#FR>8/-% &evalrG6#-%'SignumG6#&FG6#F;@)5/Fin"""3/Fin""!-FE6$&FG6#""#%'negintGC%@%FdoC$>8 (FG>8'7'Feo!""FgpFho%)infinityGC&>Fcp-%'subsopG6%/F]p%%NULLG/F;F_qFG>8%F[p>Fep- %(dev/devG6%-%'normalG6#-%(convertG6$7#-%$seqG6$*$,&FeoFeo*&8)Feo%&_XasyGFeoFgp Fgp/Fdr;Feo,$FbqFgp%"+G,&9%FeoFgpFeoFjr>Fep7&FeoFgpFgp-F_r6$-%#opG6#7$,$&Fep6#, $FdrF]pFgp,&FdrFeoFgpFeo/Fdr;FeoF[s>8+-%(dev/addG6$-F"6$7)FeoFeoFhoFeoFeoFhoFhp FjrFep-F=6#-F^t6$-%/dev/multbyrealG6$-%(dev/powG6%FcpFgpF[s&F\tF\p-%-dev/endofd evG6%FcpF[s-F\q6&/FeoF_qF^qF`qF\t/FinFgpC)>Fin-F[o6#-F^o6#-%#ReG6#-%+dev/lcoeff GFX@&FduFI/Fin%%FAILG-%&ERRORG6#Fbv>8*-%'dev/lnG6$FGF[s@$32F[s-%+dev/lengthG6#F gv0&Fgv6#-%%nopsGF`wFhp-F=F`w>Fep7#-F_r6$-Fas6#7$,$*&-%*bernoulliG6#,$FbqF]pFeo FbqFgp#FgpF]pFbq/FbqFjs>8--F[u6%FGFgpF[s-F=6#-F^t6$-F^t6$Fgv-Fht6$FgxFdx-F_u6%- F[u6%FgxF]pFjrFjrFepFgo@'/F[pFeo-F=6#-F_u6%F[qF[s7%,$%&gammaGFgpFho-F_r6$-Fas6# 7$*&)FgpFbqFeo-%%ZetaG6#FbqFeo,&FbqFeoFgpFeo/Fbq;F]p,&F[sFeoFeoFeo-FE6$F[p%(int egerGC&>Fep-Feq6%-Fhq6#-F[r6$7#-F_r6$*$,&FeoFeo*&FbqFeoFerFeoFeoFgp/Fbq;Feo,&F[ pFeoFgpFeoFirF[sF[s>Fep7$Feo-F_r6$-Fas6#7$&FepFbxFiz/Fbq;FeoF\[l>F\t-F"6$FbtF[s -F=6#-F_u6%F[qF[s-F\q6$Fcu-F^t6$FepF\tC$@%-FE6$F[pFO>FR-F"6$7%-Fas6#F[pFhoFhpF[ s>FR-F"6$F[pF[s@%32F[s-F_wFgn0&FR6#-FewFgnFhpFfnC(>Fep7$FRFho>Fdr-FW6#8&>8,Feo? (FbqFeoFeoF[s%%trueGC%>Fdr-%%diffG6$FdrFa_l>Fc_l*&Fc_lFeoFbqFgp>Fep7%-Fas6#Fep* &-%%subsG6$/Fa_lF[pFdrFeoFc_lFeoFbq>8$F[q-F=6#-F_u6%Fg`lF[sFepFcvF6F6F6F6 M6R0 I.dev/polygammaR6%%"kG%"uG%"nG6.%%factG%"iG%"xG%(exampleG%$valG%%newuG%)example 2G%"jG%&kfactG%'tomultG%%initG%$sigG6#%RCopyright~Bruno~Salvy,~INRIA~Rocquencou rt,~FranceG6"C'@$-%%typeG6$9%%*undefinedG-%'RETURNG6#F>@$4-F;6$9$%(integerG-%&E RRORG6#%%FAILG@$4-F;6$F=%%listGC$>8.-%*traperrorG6#-%$PsiG6$FFF=@%/FS%*lasterro rGF?-F@6#FS>8/-%&evalrG6#-%'SignumG6#&F=6#""$@)5/Fjn"""3/Fjn""!-F;6$&F=6#""#%'n egintGC&>8,*&)!"",&FFFgoFgoFgoFgo-%*factorialG6#FFFgo@%FioC&>8(F]p>8)-%'subsopG 6%/F_p%%NULLG/FcoFeqF=>8'-%(dev/devG6%-%'normalG6#-%(convertG6$7#-%$seqG6$*$),& FgoFgo*&8%Fgo%&_XasyGFgoFfpFgpFfp/Fjr;Fgo,$F^qFfp%"+G,&9&FgoFfpFgoF`s>Fhq7&FgoF cp,&FFFfpFfpFgo-Fdr6$-%#opG6#7$*&&Fhq6#,$FjrF_pFgoFcpFgo,&FjrFgoFfpFgo/Fjr;FgoF asC%>F^qFjo>F`qF=>Fhq7'FgoFcpFdsFjo%)infinityG>8*-F"6%FF7)FgoFgoFjoFgoFgoFjoFgt F`s-F@6#-%-dev/endofdevG6%F`qFas-Fbq6$/FgoFeq-%(dev/addG6$FhqFit/FjnFfpC(>Fjn-F \o6#-F_o6#-%#ReG6#-%+dev/lcoeffG6#F=@&FhuF?/FjnFKFH>Fcp*&)Ffp,&FFFgoFfpFgoFgo-F ip6#FjvFgo>Fhq7&FcpFF,$*&FcpFgoFFFgo#FgoF_pFgp?(FjrFgoFgoF`s%%trueGC$>Fcp,$*,Fc pFgo,(FjrF_pFFFgoFfpFgoFgo,(FjrF_pFFFgo!"#FgoFgoFjrFfp,&FjrF_pFfpFgoFfpFaw>Fhq7 %-Fhs6#Fhq*&FcpFgo-%*bernoulliGF]tFgo,&FjrF_pFFFgo-F@6#-F`u6%-%(dev/powG6%F=Ffp FasFasFhqFio@'/F]pFgoC%>FcpFhp@%/-%%iremG6$FFF_pFjoC&>Fcp,$FcpFfp>8-,$),$%#PiGF _p,&FFFgoF_pFgoFaw>Fhq7&*&FcpFgo-%%ZetaG6#FgpFgoFjo*(FhyFgoF]zFfp-%$absG6#-Fbx6 #F]zFgoFgo?(FjrF_pF_pFasFcwC%>Fcp*,FcpFgo,&FFFgoFjrFgoFgo,(FFFgoFjrFgoFfpFgoFgo FjrFfpF_tFfp>Fhy,$**FhyFgoF\zF_pFjrFfp,&FjrFgoFgoFgoFfp""%>Fhq7'F^x*&FcpFgo-Fbz 6#,(FFFgoFgoFgoFjrFgoFgoFjr*(FhyFgo,(FFFgoFjrFgoF_pFgoFfp-Ffz6#-Fbx6#F\\lFgoFc[ lC&>Fhy,$)F[zFgpFaw>Fcp,$*&FcpFgoFgpFgoFfp>Fhq7&*(FhyFgoFgpFfp-Ffz6#-FbxFczFgoF jo*&FcpFgo-FbzFizFgoFgo?(FjrF_pF_pFasFcwC%>Fhy,$**FhyFgoF\zF_pFjrFfpF_tFfpFd[l> Fcp*,FcpFgoFj[lFgoF^[lFgoFc[lFfpFjrFfp>Fhq7'F^x*(FhyFgoFj[lFfp-Ffz6#-FbxFi[lFgo Fjr*&FcpFgo-FbzF`\lFgoFc[l-F@6#-F`u6%FaqFasFhq-F;6$F]pFGC'>Fcp*&)FfpFFFgoFhpFgo >Fhq-Fjq6%-F]r6#-F`r6$7#-Fdr6$*$),&FgoFgoFirFgoFgpFfp/Fjr;Fgo,&F]pFgoFfpFgoF_sF asFas>Fhq7$Fgo-Fdr6$Fgs/Fjr;Fgo,&FasFgoFgoFgo>Fit-F"6%FFF\uFas-F@6#-F`u6%FaqFas FbuC$@%-F;6$F]pFP>FS-F"6%FF7%-Fhs6#F]pFjoFgtFas>FS-F"6%FFF]pFas@%32Fas-%+dev/le ngthGFhn0&FS6#-%%nopsGFhnFgtFgnC(>Fhq7$FSFjo>8+-FX6$FF8&>FcpFgo?(FjrFgoFgoFasFc wC%>Fbbl-%%diffG6$FbblFebl>Fcp*&FcpFgoFjrFfp>Fhq7%F^x*&-%%subsG6$/FeblF]pFbblFg oFcpFgoFjr>8$Faq-F@6#-F`u6%FgclFasFhqFHF7F7F7F7 M6R0 I-dev/LambertWR6$%"uG%"nG6-%%factG%"iG%"xG%(exampleG%#l1G%#l2G%"jG%"kG%%initG%$ sigG%$resG6#%RCopyright~Bruno~Salvy,~INRIA~Rocquencourt,~FranceG6"C&@$-%%typeG6 $9$%*undefinedG-%'RETURNG6#F<@$4-F96$F;%%listG@%/-%)zerotestG6$,$F;!""-%$expG6# FK%%trueG-F>FNC$>8,-%*traperrorG6#-%)simplifyG6$-%)LambertWG6#F;Fen@%/FS%*laste rrorG-%&ERRORG6#FS-F>F\o>8--%&evalrG6#-%'SignumG6#&F;6#""$@)/F_o"""C$@$0-%#ImG6 #-%+dev/lcoeffGFfn""!F=-F>6#-%-dev/endofdevG6%F;9%7#-%$seqG6$-%#opG6#7$*()FK,&8 %F[pFKF[pF[p)FfqFeqF[p-%*factorialG6#FfqFKFfq/Ffq;F[p,&FjpF[pF[pF[p/F_oFKC(>F_o -Fao6#-Fdo6#-%#ReGFap@&5F^r/F_oFdpF=/F_o%%FAILG-F[o6#F[s>8(-%'dev/lnG6$F;Fjp>8) -Fas6$F_sFjp@+/FjpF[p>8.-%(dev/addG6$F_s-%/dev/multbyrealG6$FdsFK/Fjp""#>Fjs-F\ t6$F[t-%(dev/prdG6$Fds-%(dev/powG6%F_sFKFjp/FjpFho>Fjs-F\t6$Fdt-Fgt6$-F\t6$-F_t 6$-Fjt6%FdsFbtFjp#F[pFbtF^t-Fjt6%F_s!"#Fjp/Fjp""%>Fjs-F\t6$F^u-Fgt6$-F\t6$-F_t6 $-Fjt6%FdsFhoFjp#FKFho-F\t6$-F_t6$Ffu#!"$FbtFds-Fjt6%F_sF_wFjp-F[o6#%0Not~imple mentedG-F>6#-%+dev/reduceG6$FjsFjpFirC$@%-F96$&F;6#FbtFD>FS-F"6$7%-F`q6#F^xFdp% )infinityGFjp>FS-F"6$F^xFjp@%32Fjp-%+dev/lengthGF\o0&FS6#-%%nopsGF\oFfxF]o@%0-F H6$,$F^xFKFLFOC(>8'7$FSFdp>8*-Fen6#8&>8+F[p?(FfqF[pF[pFjpFOC%>F^z-%%diffG6$F^zF az>Fcz*&FczF[pFfqFK>F[z7%-F`q6#F[z*&-%%subsG6$/FazF^xF^zF[pFczF[pFfq>8$-%'subso pG6%/Fbt%%NULLG/FhoF[\lF;-F>6#-Fhp6%Ff[lFjpF[zC%>Ff[l-Fjt6%-Fb[l6$/F^x,$FLFKFg[ lFhuFjp@%2Fjp"#?>F[z7#-F`q6$;F[p,&FjpFbtFbtF[p7JFKFdp*&FbtFhu-FM6#F[pFhuF[p,$Fd ]l#F[vFhoFbt,$*&FbtFhuFd]l#FhoFbt#"#6"#OFho,$*$Fd]lFbt#!#V"$N"F]v,$*&FbtFhuFd]l #""&Fbt#"$p("%?VFf^l,$*$Fd]lFho#!%o<"%0&)""',$*&FbtFhuFd]l#""(Fbt#"'j3o"(+KW&Fc _l,$*$Fd]lF]v#!%ER"&:b#""),$*&FbtFhuFd]l#""*Fbt#"*dvGE#"++CY^BF``l,$*$Fd]lFf^l# !)wa5B"*v)[%*="#5,$*&FbtFhuFd]l#F\^lFbt#"-(>j%4(p"".+wD^F<#F\^l,$*$Fd]lF__l#!+E E-PA",v$=b;A"#7,$*&FbtFhuFd]l#"#8Fbt#"0rn"\;uym"2+!ob)eZo,"Fial,$*$Fd]lFc_l#!+o "4%3!)",vr<&4$*"#9,$*&FbtFhuFd]l#"#:Fbt#"3([(4DULjO5"4++C-%fcKI=Ffbl,$*$Fd]lF\` l#!0eZ1NO!Q$*"2DJIWo6jC""#;,$*&FbtFhuFd]l#"#F\sF]\lF\sF5F5F5F5 M6R0 I,dev/BesselJR6%%#nuG%"uG%"nG60%%factG%"iG%"xG%(exampleG%&sinusG%(cosinusG%"kG% "jG%%initG%%invuG%'newresG%$resG%$sigG%'tomultG6#%RCopyright~Bruno~Salvy,~INRIA ~Rocquencourt,~FranceG6"C'@$-%%typeG6$9%%*undefinedG-%'RETURNG6#F@@$-F=6$9$%%li stG-%&ERRORG6#%0Not~implementedG@$4-F=6$F?FHC$>8,-%*traperrorG6#-%(BesselJG6$FG F?@%/FS%*lasterrorGFA-FB6#FS>80-%&evalrG6#-%'SignumG6#&F?6#""$@)/Fjn"""C&>8+*&- %&GAMMAG6#FG!""FGF^p>8'7$Fio""!?(8%FfoFfo9&%%trueGC$>Fio,$*(FioFfoFdpF^p,&FGFfo FdpFfoF^p#F^p""%>F`p7%-%#opG6#F`pFioFdp-FB6#-%/dev/multbyrealG6$-%(dev/prdG6$-% (dev/powG6%F?FGFep-%-dev/endofdevG6%-F\r6%F?""#FepFepF`p*$)FcrFGF^p/FjnF^pC/>Fj n-F\o6#-F_o6#-%#ReG6#-%+dev/lcoeffG6#F?@&FfrFA/Fjn%%FAILG-FJ6#Fes>8(-%+dev/coss inG6$-%(dev/addG6$,&*&FGFfo%#PiGFfo#F^pFcrFbtF\qF?Fep>8)&Fis6#Ffo>8--F\r6%F?F^p Fep>Fis-Fiq6$&Fis6#FcrFit>8$-F\r6%FitFcrFep>81Ffo>8/Fbp>8.-F^t6$Fet-Ffq6$Fis,&* $FGFcrFct#Ffo"")Ffo?(FdpFfoFfo,&-%&truncG6#,$Fep#FfoFcrFfoFfoFfo0FhuFjuC%>FhuFj u>Ffu-Fiq6$FfuFbu>Fju-F^t6$Fhu-F^t6$-Ffq6$-Fiq6$FetFfu**)F^pFdpFfo-%(productG6$ ,&F`vF]q*$,&.8*FcrF^pFfoFcrF^p/Fax;Ffo,$FdpFcrFfo)FbvFexF^p-%*factorialG6#FexF^ p-Ffq6$-Fiq6$FisFfu**)F^p,&FdpFfoFfoFfoFfo-F\x6$F^x/Fax;Ffo,&FdpFcrFfoFfoFfo)Fb vFeyF^p-Fhx6#FeyF^p>Fhu-Ffq6$-Fiq6$Fju-F\r6%FitFivFep*&FcrFiv*$FbtF^pFiv@'32Fep -%+dev/lengthG6#Fhu0&Fhu6#-%%nopsGFgz%)infinityG-FBFgz/FizF][l-FB6#7#-Faq6$;Ffo ,&F[[lFfo!"#FfoFhuF^[l/FjnFbpC$@%-F=6$&F?F`uFH>FS-F"6%FG7%-Faq6#F]\lFbpF][lFep> FS-F"6%FGF]\lFep@%32Fep-FfzFhn0&FS6#-F\[lFhnF][lFgnC(>F`p7$FSFbp>Fio-FX6$FG8&>F bxFfo?(FdpFfoFfoFepFfpC%>Fio-%%diffG6$FioFe]l>Fbx*&FbxFfoFdpF^p>F`p7%F`q*&-%%su bsG6$/Fe]lF]\lFioFfoFbxFfoFdp>Fbu-%'subsopG6%/Fcr%%NULLG/FcoF[_lF?-FB6#-F_r6%Fb uFepF`pFfsF9F9F9F9 M6R0 I+dev/AiryAiR6$%"uG%"nG68%%factG%"iG%"xG%(exampleG%(expfactG%(cosinusG%&sinusG% &fact2G%"kG%#c1G%#c2G%#ckG%)example1G%)example2G%"jG%&kfactG%%num1G%%num2G%%coe fG%%initG%$sigG%%zetaG6#%RCopyright~Bruno~Salvy,~INRIA~Rocquencourt,~FranceG6"C &@$-%%typeG6$9$%*undefinedG-%'RETURNG6#FG@$4-FD6$FF%%listGC$>87-%*traperrorG6#- %'AiryAiG6#FF@%/FR%*lasterrorGFH-FI6#FR>88-%&evalrG6#-%'SignumG6#&FF6#""$@)/Fin """C*>8-,$*&Fbo#FeoFbo-%&GAMMAG6##""#Fbo!""F[p>8.,$*&FboF_p-F]p6#F[pFap#FapFbo> 84Feo>85Feo>83Feo>8'7&Fho""!FcpFeo?(8%FeoFeo,&9%FeoFapFeo%%trueGC&>F^q,$**F^qFe o,&FdqFbo!"#FeoFeo,&FdqFboFapFeoFeoFdqFeoFbo>Fjp*&FjpFeoF\rFeo>F\q*&F\qFeoF^rFe o>F`q7'-%#opG6#F`q*(FhoFeoFjpFeoF^qFap,$FdqFbo**FcpFeoF\qFeoF^qFap,&FdqFboFeoFe oFapF[s-FI6#-%-dev/endofdevG6%FFFfqF`q/FinFapC%>86-%&evalcG6#-%+dev/lcoeffGFX>F in-F[o6#-F^o6#-%&coeffG6%Fds^#FeoFbq@'/Fin%%FAILG-%&ERRORG6#Fet3Fas/-F`t6%FdsFb tFeoFbqC->89-%/dev/multbyrealG6$-%(dev/powG6%FF#FboF`pFfqF_p>8$-Fdu6%F_uFap,&Ff qFeoFeoFeo>8+-Fdu6%FhuF`pF[v>8)-%+dev/cossinG6$-%(dev/addG6$,$%#PiG#Feo""%F]vFf q>8*&Fav6#F`p>Fav-Fau6$-%(dev/prdG6$&Fav6#FeoFhuFap>807$FeoFeo>817$#""&"#sFeo>8 /F^x?(FdqFeoFeo-%%iquoG6$FeqF`pFgqC&>Fbx,$**FbxFeo,&Fdq"#7FapFeoFeo,&FdqF\y!"&F eoFeoFdqFap#Fap"$W">Fiw7%-Ffr6#FiwFbxFdq>Fbx,$**FbxFeo,&FdqF\yF_xFeoFeo,&FdqF\y FeoFeoFeo,&FdqF`pFeoFeoFap#FeoF`x>F\x7%-Ffr6#F\xFbxFdq-FI6#-Fau6$-Fdw6$-Fdu6%FF #FapF[wFfq-Ffv6$-Fdw6$F]w-F_s6%F]v-Fex6$F[vF`pFiw-Fdw6$Fav-F_s6%F]vF_[lF\x*$Fiv #FapF`pC%>FhuF`u>8(-%(dev/expG6$-Fau6$FhuFapF[v@%32Ffq-%+dev/lengthG6#Fj[l0&Fj[ l6#-%%nopsGFe\l%)infinityG-FI6#-%+dev/reduceG6$-Fau6$-Fdw6$FfzFj[l,$*$Fiv#FeoF` pFg]lFfqC'>Fhu-Fdu6%FhuFapFfq>F`q7$FeoFbq>FbxFeo?(FdqFeoFeoFfqFgqC$>Fbx,$**FbxF eo,&Fdq""'F^yFeoFeo,&FdqFe^lFapFeoFeoFdqFap#FapF`x>F`q7%FerFbxFdq-FI6#-F_]l6$-F au6$-Fdw6$-Fdw6$Ffz-F_s6%FhuFfqF`qFj[l,$Fe[lFg]lFfq/FinFbqC$@%-FD6$&FFF_wFO>FR- F"6$7%-Ffr6#F\`lFbqF[]lFfq>FR-F"6$F\`lFfq@%32Ffq-Fd\lFgn0&FR6#-Fj\lFgnF[]lFfnC( >F`q7$FRFbq>82-FW6#8&>8,Feo?(FdqFeoFeoFfqFgqC%>Fbal-%%diffG6$FbalFeal>Fgal*&Fga lFeoFdqFap>F`q7%Fer*&-%%subsG6$/FealF\`lFbalFeoFgalFeoFdq>Fhu-%'subsopG6%/F`p%% NULLG/FboF\clFF-FI6#Fd_lFftF@F@F@F@ M6R0 I*dev/dilogR6$%"uG%"nG6.%%factG%"iG%"xG%(exampleG%&itslnG%"jG%"kG%%initG%'newre sG%$resG%$sigG%'tomultG6#%RCopyright~Bruno~Salvy,~INRIA~Rocquencourt,~FranceG6" C&@$-%%typeG6$9$%*undefinedG-%'RETURNG6#F=@$4-F:6$F<%%listGC$>8+-%*traperrorG6# -%&dilogG6#F<@%/FH%*lasterrorGF>-F?6#FH>8.-%&evalrG6#-%'SignumG6#&F<6#""$@)/FU" ""C*-%+dev/lcoeffGFN>8$F<>8(-%'dev/lnG6$F<9%>8/F[o>8-""!>8,,$*$%#PiG""##F[o""'? (8%F[oF[oFfo0FjoF]pC%>FjoF]p>Fho-%(dev/prdG6$FhoF`o>F]p-%(dev/addG6$Fjo-F[q6$Fh o-F_q6$,$*$Fep!"#!""-%/dev/multbyrealG6$Fbo*$FepFhq@'32Ffo-%+dev/lengthG6#F]p0& F]p6#-%%nopsGFbr%)infinityG-F?Fbr/-%#opG6$FfrF]pFhr-F?6#7#-F\s6$;F[o,&FfrF[oFgq F[oF]pFir/FUFhqC+>FU-FW6#-FZ6#-%#ReG6#F]o@&5Fes/FUF[pF>3/FU%%FAILG-F:6$&F<6#Fap %)constantG-%&ERRORG6#Fdt>F`o-%(dev/powG6%FFboFco>FhoF[o>FjoF[p>F]p-F_q 6$,$F_p#FhqFcp-Fjq6$-F_u6%FboFapFfo#FhqFap?(FepF[oF[oFfoFfpC%>FjoF]p>FhoFjp>F]p -F_q6$Fjo-F[q6$Fho-F_q6$FfqFiq@'F^rFir/FdrFhrF^sFirFatC$@%-F:6$FgtFE>FH-F"6$7%- F\s6#FgtF[pFhrFfo>FH-F"6$FgtFfo@%32Ffo-FarFS0&FH6#-FgrFSFhrFRC(>8'7$FHF[p>8)-FM 6#8&>8*F[o?(FepF[oF[oFfo%%trueGC%>Fex-%%diffG6$FexFhx>Fjx*&FjxF[oFepFhq>Fbx7%-F \s6#Fbx*&-%%subsG6$/FhxFgtFexF[oFjxF[oFep>F`o-%'subsopG6%/Fap%%NULLG/FhnFbzF<-F ?6#-%-dev/endofdevG6%F`oFfoFbxFjtF6F6F6F6 M6R0 I(dev/erfR6$%"uG%"nG6.%%factG%"iG%"xG%(exampleG%%sideG%"kG%#ckG%"jG%%coefG%%ini tG%$resG%$sigG6#%RCopyright~Bruno~Salvy,~INRIA~Rocquencourt,~FranceG6"C&@$-%%ty peG6$9$%*undefinedG-%'RETURNG6#F=@$4-F:6$F<%%listG-F?6#-%$erfG6#F<>8/-%&evalrG6 #-%'SignumG6#&F<6#""$@)/FL"""-F?6#-%(dev/prdG6$F<-%-dev/endofdevG6%-%(dev/powG6 %F<""#9%F_o7#-%$seqG6$-%#opG6#7$,$**)!""8%FX-%*factorialG6#F\pF[p,&F\pF^oFXFXF[ p-%%sqrtG6#%#PiGF[pF^oF\p/F\p;""!,&F_oFXFXFX/FLF[pC->8,-%&evalcG6#-%+dev/lcoeff GFJ@$0-%&coeffG6%F\q^#FXFXFgp-%&ERRORG6#%%FAILG>8(-FN6#-FQ6#F\q@$30F]rFX0F]rF[p Fhq>8$-F\o6%F8'7&FXFgp#F[pF^oFX>8*F^s?(F\pF^oFXF_o%%trueGC$>F`s,$*&F`sF X,&F\pF^oF[pFXFXF^s>F\s7%-Feo6#F\sF`sF\p>8.-Ffn6$-F\o6%FF]t-%(dev/addG6$FX-%/dev/multbyrealG6$-Ffn6$-%(dev/expG6$-F[u6$F[oF[pF_ oF]t,$*$FdpF^sF[p>F]t-Fht6$F[p-F[u6$F]uFeu@'32F_o-%+dev/lengthG6#F]t0&F]t6#-%%n opsGF`v%)infinityG-F?F`v/-Feo6$FdvF]tFfv-F?6#7#-Feo6$;FX,&FdvFXFjrFXF]tFgv/FLFg pC$@%-F:6$&F<6#F^oFE>8--F"6$7%-Feo6#FgwFgpFfvF_o>Fjw-F"6$FgwF_o@%32F_o-F_v6#Fjw 0&Fjw6#-FevFgxFfv-F?FgxC(>F\s7$FjwFgp>8+-FI6#8&>8)FX?(F\pFXFXF_oFbsC%>Fay-%%dif fG6$FayFdy>Ffy*&FfyFXF\pF[p>F\s7%Fjs*&-%%subsG6$/FdyFgwFayFXFfyFXF\p>Fgr-%'subs opG6%/F^o%%NULLG/FUF[[lF<-F?6#FbtFhqF6F6F6F6 M6R0 I)dev/ZetaR6$%"uG%"nG6(%$sigG%$resG%"iG%'oldresG%%initG%(exampleG6#%RCopyright~ Bruno~Salvy,~INRIA~Rocquencourt,~FranceG6"C'@$/9#""$-%&ERRORG6#%4not~implemente d~yetG@$-%%typeG6$9$%*undefinedG-%'RETURNG6#F?@$4-F<6$F>%%listG@'/-%)zerotestG6 $F>"""%%trueGF@/-FK6#-%%ZetaG6#F>FN-FA6#""!-FAFQ>8$-%&evalrG6#-%'SignumG6#&F>6# F5@)/FZFM-FA6#-%-dev/endofdevG6%F>9%7%-FSFVFW-%$seqG6$-%#opG6#7$*&-FS6$8&FWFM-% *factorialG6#Fap!""Fap/Fap;FMFdo/FZFWC%@%-F<6$&F>6#""#FG>8(-F"6$7%-F[p6#F]qFW%) infinityGFdo>Faq-F"6$F]qFdo@$32Fdo-%+dev/lengthG6#Faq0&Faq6#-%%nopsGF`rFgq-FAF` r@%4-F<6$FaqF?C$@%/FaqFW>8)7#-Fho6$-F[p6#7$*&-FS6$FapF]qFMFbpFepFap/Fap;FM,&Fdo FMFMFM>F_s7%FaqFW-Fho6$FcsFfp-FA6#-Fbo6%-%'subsopG6%/F_q%%NULLG/F5FhtF>FdoF_sC$ >F_s7%%&gammaGFW-Fho6$-F[p6#7$*()FepFapFM-F]uFdpFMFbpFepFap/Fap;FM-%$maxG6$FM,& FdoFM!"#FM-FA6#-%(dev/addG6$-%(dev/powG6%FdtFepFdo-Fbo6%Fdt,&FdoFMFepFMF_s/FZFe pC$>FZ-Ffn6#-Fin6#-%#ReG6#-%+dev/lcoeffGFT@)Fhp-F76#%%FAILGF^oC'>8'FW>8%FM?(Fap F_qFMF[t0F\xFjwC$>FjwF\x>F\x-F`v6$F\x-%(dev/expG6$-%/dev/multbyrealG6$F>,$-%#ln GFdpFepFdo@$/&F\x6#-Fer6#F\xFgq>F\x7#-F[p6$;FM,&FayFMF\vFMF\x-FAFbyFhvC$>F\x-F` v6$FM-Fhx6$F>Fep-FA6#-%(dev/prdG6$-Fhx6$-Fcz6$-Fex6$-Fhx6$F>-F\y6#,$%#PiGF_qFdo -Fcz6$-%*dev/GAMMAG6$F\xFdo-F"Fe[l*$F`[lFep-%(dev/sinG6$-Fhx6$F>,$F`[l#FMF_qFdo FewFewF0F0F0F0 M6R0 I+dev/cossinR6$%"uG%"nG60%"iG%#i0G%'inivalG%%newuG%$resG%#coG%#siG%'still0G%%fa ctG%&interG%(infpartG%'newresG%$auxG%%nbitG6#%RCopyright~Bruno~Salvy,~INRIA~Roc quencourt,~FranceG6"@)-%%typeG6$9$%*undefinedGF>4-F;6$F=%%listG7$-%$cosG6#F=-%$ sinGFF/9%""!7$"""FKC$>80-%&evalrG6#-%'SignumG6#&F=6#""$@'5/FPFM/FPFKC*>8(7"@'Fg nC%@%4-F;6$&F=6#""#FB>8&-F"6#Fbo>Ffo-F"6$7%-%#opGFhoFK%)infinityGFJ>8+-%'member G6$FKFfo>8'-%'subsopG6%/Fdo%%NULLG/FYF[qF=F_o@%2FY-%%nopsGFFC)>8)7%&F=6#FMFMFK> 8*7%FeqFboFW>8,Fbo?(8$FdoFdo,$FJFdo%%trueGC&>F[r,$*(F[rFMF]r!""FboFMFdr>Fcq7%-F ^p6#FcqF[r*&F]rFMFWFM>F[r*(F[rFM,&F]rFMFMFMFdrFboFM>Fhq7%-F^p6#FhqF[r*&F\sFMFWF M>Ffo7$FcqFhq>Fap%&falseG>FfpFgp-%'RETURNG6#7$7'FeqFMFKFM,$FWFdoF=C%>FfoFL>FfpF =>FapF_r@$3Fap2FY-F`q6#FfpC%>8--F"6$7'&FfpFfq&FfpFco&FfpFXFKF_p-%&truncG6#-%&ev alfG6#*&,&FJFMFMFMFMF^uFdr@*/FfoFL>FfoFht/Ffo7$FKFM>Ffo7$-%/dev/multbyrealG6$&F htFcoFdr&FhtFfq/Ffo7$FdrFK>Ffo7$-F_v6$FbvFdrF^v/Ffo7$FKFdr>Ffo7$FavFgv>Ffp-Fhp6 %FjpF\qFfp>8%F^u@%/&Ffp6#FdtF_p>81,&FdtFM!"$FM>Fgw,&FdtFMFdrFM>8/Ffo?(F]rFdoFdo Fgw0F]xFjnC%>FjnF]x>Fht-F"6$7'F\u&Ffp6#F]r&Ffp6#F\sFKF_p-F`u6#-Fcu6#,$*(FfuFMFa wFMFhxFdr#FMFdo>F]x7$-%(dev/addG6$-%(dev/prdG6$&FjnFfqFbv-F_v6$-Fgy6$&FjnFcoFav Fdr-Fdy6$-Fgy6$FiyFav-Fgy6$F^zFbv@%/FjnF[o-Fhs6#Ffo-Fhs6#F]x/FPFdrC%?(F]r""&FMF _q-Fcp6$-FR6#-FU6#&F=Fgx<$FdrFKF8>8.-%,dev/instancG6#7#-F^p6$;FM,&F]rFM!"#FMF=@ %31F]rF_q0Fe[lF_pC$>Fht-F"6$7$Feq-F^p6$;,&F]rFMFdrFMF_qF=FJ-Fhs6#7$-Fdy6$-F_v6$ Fbv-FE6#Fh[l-F_v6$Fav,$-FHFg]lFdr-Fdy6$-F_v6$FbvF[^l-F_v6$FavFf]l-Fhs6#7$Ff]lF[ ^l-%&ERRORG6#%%FAILGF8F8F8F8 M6R0 I(dev/cosR6$%"uG%"nG6"6#%RCopyright~Bruno~Salvy,~INRIA~Rocquencourt,~FranceGF'- %'RETURNG6#&-%+dev/cossinG6$9$9%6#"""F'F'F'F' M6R0 I(dev/sinR6$%"uG%"nG6"6#%RCopyright~Bruno~Salvy,~INRIA~Rocquencourt,~FranceGF'- %'RETURNG6#&-%+dev/cossinG6$9$9%6#""#F'F'F'F' M6R0 I(dev/tanR6$%"uG%"nG6*%%factG%"iG%(exampleG%&exponG%&kfactG%&interG%%initG%$sig G6#%RCopyright~Bruno~Salvy,~INRIA~Rocquencourt,~FranceG6"C&@$-%%typeG6$9$%*unde finedG-%'RETURNG6#F9@$4-F66$F8%%listGC$>8*-%*traperrorG6#-%$tanG6#F8@'5/FD%*las terrorG-%$hasG6$FD%)infinityGF:/-%)zerotestG6#FD%%trueG-F;6#""!-F;FV>8+-%&evalr G6#-%'SignumG6#&F86#""$@)/Fgn"""C(>8$-%(dev/powG6%F8""#9%>8&7*FcoFZ#FcoF`oFco#F jo"#:Fjo#"#<"$:$F`o>8'"$c#>8(!&?.%?(8%""&Fco,&F[pFcoFcoFcoFWC%>Ffp,$Ffp""%>Fip, $*(FipFcoF\qFco,&F\qFjo!""FcoFco!"#>F]p7%-%#opG6#F]p**FfpFco,&FfpFcoFgqFcoFcoFi pFgq-%*bernoulliG6#,$F\qFjoFco,&F\qFcoFgqFco-F;6#-%(dev/prdG6$F8-%-dev/endofdev G6%FfoF[pF]p/FgnFgqF:/FgnFZC$@%-F66$&F86#FjoFA>FD-F"6$7%-F\r6#FcsFZFRF[p>FD-F"6 $FcsF[p@'-F66$FDF9-Fho6%-%(dev/cotG6$F8F[pFgqF[p32F[p-%+dev/lengthGFV0&FD6#-%%n opsGFVFRFenC$>8)-F"6$-%'subsopG6%/Fjo%%NULLG/F`oFhuF8F[p-F;6#-Fhr6$-%(dev/addG6 $FauFD-Fho6%-F_v6$Fco-Fhr6$Fau-%/dev/multbyrealG6$FDFgqFgqF[p-%&ERRORG6#%%FAILG F2F2F2F2 M6R0 I(dev/cotR6$%"uG%"nG6)%%factG%"iG%(exampleG%&kfactG%%initG%&interG%$sigG6#%RCop yright~Bruno~Salvy,~INRIA~Rocquencourt,~FranceG6"C%@$4-%%typeG6$9$%%listGC$>8(- %*traperrorG6#-%$cotG6#F8@'5/F<%*lasterrorG-%$hasG6$F<%)infinityG-%'RETURNG6#%* undefinedG/-%)zerotestG6#F<%%trueG-FL6#""!-FLFR>8*-%&evalrG6#-%'SignumG6#&F86#" "$@)/FY"""C'>8$-%(dev/powG6%F8""#9%>8&7(#!""F\oFV#F\p"#XF_o#!"#"$X*Ffo>8'#!"%F^ p?(8%""%F_oFgoFSC$>Fcp,$*(FcpF_oFgpF\p,&FgpFfoF\pF_oF\pF`p>Fio7%-%#opG6#Fio*&Fc pF_o-%*bernoulliG6#,$FgpFfoF_o,&FgpF_oF\pF_o-FL6#-%(dev/addG6$-Fdo6%F8F\pFgo-%( dev/prdG6$F8-%-dev/endofdevG6%Fbo,&FgoF_oF\pF_oFio/FYF\pFK/FYFVC$@%-F66$&F86#Ff oF9>F<-F"6$7%-Faq6#F]sFVFJFgo>F<-F"6$F]sFgo@'-F66$F8)-F"6$-%'subsopG6%/Ffo%%NULLG /F\oFbuF8Fgo-FL6#-Far6$-F\r6$-Far6$F8(-%*traperrorG6#-%$secG6#F6@%5/FB%*lasterrorG-%$hasG6$ FB%)infinityGF8-F96#FB>8)-%&evalrG6#-%'SignumG6#&F66#""$@)/FT"""C'>8$-%(dev/pow G6%F6""#9%>8&7*Fjn""!#FjnFaoFjn#""&"#CFao#"#h"$?(Fgn>8'!$?(?(8%""%FjnFbo%%trueG C$>F_p,$*(F_pFjnFbpFjn,&FbpFao!""FjnFjn!"#>Fdo7%-%#opG6#Fdo*&-%&eulerG6#,$FbpFa oFjnF_pFjpFbp-F96#-%-dev/endofdevG6%F]oFboFdo/FTFjpF8/FTFfo-F96#-F_o6%&-%+dev/c ossinG6$F6Fbo6#FjnFjpFbo-%&ERRORG6#%%FAILGF0F0F0F0 M6R0 I(dev/cscR6$%"uG%"nG6)%%factG%"iG%%initG%'newresG%$resG%'tomultG%$sigG6#%RCopyr ight~Bruno~Salvy,~INRIA~Rocquencourt,~FranceG6"C%@$4-%%typeG6$9$%%listGC$>8&-%* traperrorG6#-%$cscG6#F8@%5/F<%*lasterrorG-%$hasG6$F<%)infinityG-%'RETURNG6#%*un definedG-FL6#F<>8*-%&evalrG6#-%'SignumG6#&F86#""$@)/FR"""C(>8$-%(dev/powG6%F8"" #9%>8)F8>8(""!>8'-%(dev/addG6$-F]o6%F8!""F`o-%/dev/multbyrealG6$F8#Fhn""'?(8%F_ oFhn,&F`oFhnFhnFhn0FgoFdoC%>FdoFgo>Fbo-%(dev/prdG6$FboF[o>Fgo-Fio6$Fdo-F_p6$Fbo ,$**)F]p,&FdpFhnF]pFhnFhn,&)F_o,&FdpF_oF]pFhnFhnF]pFhnFhn-%*bernoulliG6#,$FdpF_ oFhn-%*factorialGF[rF]pF_o-FL6#-%+dev/reduceG6$FgoF`o/FRF]pFK/FRFeo-FL6#-F]o6%& -%+dev/cossinG6$F8F`o6#F_oF]pF`o-%&ERRORG6#%%FAILGF1F1F1F1 M6R0 I+dev/arccosR6$%"uG%"nG6"6#%RCopyright~Bruno~Salvy,~INRIA~Rocquencourt,~FranceG F'-%(dev/addG6$,$%#PiG#"""""#-%/dev/multbyrealG6$-%+dev/arcsinG6$9$9%!""F'F'F'F ' M6R0 I+dev/arcsinR6$%"uG%"nG6+%%factG%"iG%"kG%"xG%(exampleG%"jG%&kfactG%%initG%$sigG 6#%RCopyright~Bruno~Salvy,~INRIA~Rocquencourt,~FranceG6"C&@$-%%typeG6$9$%*undef inedG-%'RETURNG6#F:@$4-F76$F9%%listG-F<6#-%'arcsinG6#F9>8,-%&evalrG6#-%'SignumG 6#&F96#""$@)/FI"""C'>8$-%(dev/powG6%F9""#9%>8(7(FU""!#FU""'FU#FR"#SFfn>8*#FR"") ?(8%FRFUFgn%%trueGC$>Fao,$*(FaoFU,&FeoFfn!""FUFUFeoF\p#FUFfn>Fin7%-%#opG6#Fin*& FaoFU,&FeoFfnFUFUF\pFeo-F<6#-%(dev/prdG6$F9-%-dev/endofdevG6%FXFgnFin/FIF\pF;/F IF[oC'@%-F76$&F96#FfnFB>8+-F"6$7%-Fap6#FcqF[o%)infinityGFgn>Ffq-F"6$FcqFgn@$32F gn-%+dev/lengthG6#Ffq0&Ffq6#-%%nopsGFerF\r-FFI-FK6#-FN6#,&FcqFUF\pFU@)FTF;F^qC'>FX-%'subsopG6%/Ffn%%NULLG/FRFbtF9 >Fin7&FUF[o#F\p"#7FU>Fao#F\p""%?(FeoFfnFU,&FgnFUF\pFUFfoC$>Fao,$FjoFit>FinF_p-F <6#-%(dev/addG6$,$%#PiGF]p-%/dev/multbyrealG6$-Fhp6$-FZ6%-Fiu6$FXF\pF]pF\u-F[q6 %FXF\uFin,$*$FfnF]pF\pF]qC$>FI-FK6#-FN6#,&F\pFUFcqF\p@)FTF;F^qC'>FXF^t>Fin7&FUF [o#FUFgtFU>Fao#FUFjt?(FeoFfnFUF\uFfoC$>Fao,$FjoFcw>FinF_p-F<6#-Fdu6$,$Fgu#F\pFf n-Fiu6$-Fhp6$-FZ6%FXF]pF\uFavFcvF]qC$>FfqF^r@%FarF[sC(>Fin7$FfqF[o>8)-FF6#8'>8& FU?(FeoFUFUFgnFfoC%>F\y-%%diffG6$F\yF_y>Fay*&FayFUFeoF\p>Fin7%F`p*&-%%subsG6$/F _yFcqF\yFUFayFUFeo>FXF^t-F<6#FjpFasFasFasF3F3F3F3 M6R0 I+dev/arctanR6$%"uG%"nG6+%%factG%"iG%"xG%(exampleG%%invuG%%initG%"jG%"kG%$sigG6 #%RCopyright~Bruno~Salvy,~INRIA~Rocquencourt,~FranceG6"C&@$-%%typeG6$9$%*undefi nedG-%'RETURNG6#F:@$4-F76$F9%%listG-F<6#-%'arctanG6#F9>8,-%&evalrG6#-%'SignumG6 #&F96#""$@)/FI"""C$>8$-%(dev/powG6%F9""#9%-F<6#-%(dev/prdG6$F9-%-dev/endofdevG6 %FXFgn7#-%$seqG6$-%#opG6#7$*&)!""8%FU,&F[pFfnFUFUFjoF[p/F[p;""!,&FgnFUFUFU/FIFj oC$>FI-%)dev/signG6#&F96#Ffn@)/FI%%FAILG-%&ERRORG6#F[qFTC$>FX-FZ6%F9!"#,&FgnFUF joFU-F<6#-%(dev/addG6$,$%#PiG#FUFfn-F[o6$-FZ6%F9FjoFdq-F^o6%FXFdq7#-Fbo6$-Feo6# 7$*&)Fjo,&F[pFUFUFUFUF\pFjoF[p/F[p;F_pFdqFapC%>8(F_r>FX-FZ6%F`sFfnFdq-F<6#-Fhq6 $,$F[r#FjoFfn-F[o6$F`s-F^o6%FXFgn7#-Fbo6$Fdo/F[p;F_pFgnF\q/FIF_pC$@%-F76$FgpFB> 8)-F"6$7%-FeoFfpF_p%)infinityGFgn>Fit-F"6$FgpFgn@%32Fgn-%+dev/lengthG6#Fit0&Fit 6#-%%nopsGFguF^u-F8'7$FitF_p>8*-FF6#8&>8+FU?(F[pFUFUFgn%%trueGC%>Fcv-%%d iffG6$FcvFfv>Fhv*&FhvFUF[pFjo>F`v7%-Feo6#F`v*&-%%subsG6$/FfvFgpFcvFUFhvFUF[p>FX -%'subsopG6%/Ffn%%NULLG/FRF`xF9-F<6#-F^o6%FXFgnF`vF\qF3F3F3F3 M6R0 I+dev/arccotR6$%"uG%"nG6"6#%RCopyright~Bruno~Salvy,~INRIA~Rocquencourt,~FranceG F'-%(dev/addG6$,$%#PiG#"""""#-%/dev/multbyrealG6$-%+dev/arctanG6$9$9%!""F'F'F'F ' M6R0 I+dev/arccscR6$%"uG%"nG6(%%factG%"iG%(exampleG%&kfactG%%initG%$sigG6#%RCopyrigh t~Bruno~Salvy,~INRIA~Rocquencourt,~FranceG6"C&@$-%%typeG6$9$%*undefinedG-%'RETU RNG6#F7@$4-F46$F6%%listGC$>8(-%*traperrorG6#-%'arccscG6#F6@%5/FB%*lasterrorG-%$ hasG6$FB%)infinityGF8-F96#FB>8)-%&evalrG6#-%'SignumG6#&F66#""$@)/FT!""C'>8$-%(d ev/powG6%F6!"#9%>8&7&"""""!#Ffo""'Ffo>8'#Ffo""#?(8%F]pFfoFbo%%trueGC$>F[p,$*(F[ pFfo,&F_pF]pFjnFfoFfoF_pFjnF\p>Fdo7%-%#opG6#Fdo*&F[pFfo,&F_pF]pFfoFfoFjnF_p-F96 #-%(dev/prdG6$-F_o6%F6FjnFbo-%-dev/endofdevG6%F]oFboFdo/FTFfoF8/FTFgo-F96#-%+de v/arcsinG6$FbqFbo-%&ERRORG6#%%FAILGF0F0F0F0 M6R0 I+dev/arcsecR6$%"uG%"nG6"6#%RCopyright~Bruno~Salvy,~INRIA~Rocquencourt,~FranceG F'-%(dev/addG6$,$%#PiG#"""""#-%/dev/multbyrealG6$-%+dev/arccscG6$9$-%$maxG6$F1, &9%F0!""F0F>F'F'F'F' M6R0 I)dev/sinhR6$%"uG%"nG6(%%factG%"iG%(exampleG%&ifactG%%initG%$sigG6#%RCopyright~ Bruno~Salvy,~INRIA~Rocquencourt,~FranceG6"C&@$-%%typeG6$9$%*undefinedG-%'RETURN G6#F7@$4-F46$F6%%listG-F96#-%%sinhG6#F6>8)-%&evalrG6#-%'SignumG6#&F66#""$@)/FF" ""C'>8$-%(dev/powG6%F6""#9%>8&7(FR""!#FR""'FR#FR"$?"FY>8'F\o?(8%""%FR,&FZFRFRFR %%trueGC$>F^o*(F^oFR,&F`oFR!""FRFRF`oFR>Ffn7%-%#opG6#Ffn*$F^oFhoFgo-F96#-%(dev/ prdG6$F6-%-dev/endofdevG6%FUFZFfn/FFFhoC$>FF-FH6#-FK6#-%+dev/lcoeffGFD@'FQC$>8( -%(dev/expG6$F6FZ@%32FZ-%+dev/lengthG6#Fcq0&Fcq6#-%%nopsGF\r%)infinityG-F96#-%/ dev/multbyrealG6$Fcq#FRFY-F96#-Ffr6$-%(dev/addG6$Fcq-Ffr6$-FW6%FcqFhoFZFhoFhrFg pC$>Fcq-Feq6$-Ffr6$F6FhoFZ@%Fhq-F96#-Ffr6$Fcq#FhoFY-F96#-Ffr6$-F^s6$-Ffr6$FcqFh oFbsFhr-%&ERRORG6#%%FAILG/FFFhn-F96#-F^s6$-Fbp6$-F"6$&F66#FYFZ-%)dev/coshG6$-%' subsopG6%/FY%%NULLG/FOF^vF6FZ-Fbp6$-FhuFdu-F"FiuFhtF0F0F0F0 M6R0 I)dev/coshR6$%"uG%"nG6(%%factG%"iG%&ifactG%(exampleG%%initG%$sigG6#%RCopyright~ Bruno~Salvy,~INRIA~Rocquencourt,~FranceG6"C&@$-%%typeG6$9$%*undefinedG-%'RETURN G6#F7@$4-F46$F6%%listG-F96#-%%coshG6#F6>8)-%&evalrG6#-%'SignumG6#&F66#""$@)/FF" ""C'>8$-%(dev/powG6%F6""#9%>8'7(FR""!#FRFYFR#FR"#CFY>8&F[o?(8%FOFRFZ%%trueGC$>F ]o,$*(F]oFR,&F_oFY!""FRFRF_oFRFY>Ffn7%-%#opG6#Ffn*$F]oFfoF_o-F96#-%-dev/endofde vG6%FUFZFfn/FFFfoC$>FF-FH6#-FK6#-%+dev/lcoeffGFD@&FQC$>8(-%(dev/expG6$F6FZ@%32F Z-%+dev/lengthG6#F^q0&F^q6#-%%nopsGFgq%)infinityG-F96#-%/dev/multbyrealG6$F^qFi n-F96#-Far6$-%(dev/addG6$F^q-FW6%F^qFfoFZFinFbpC$>F^q-F`q6$-Far6$F6FfoFZ@%FcqF^ r-F96#-Far6$-Fhr6$F^q-FW6$F^qFfoFin/FFFhn-F96#-Fhr6$-%(dev/prdG6$-F"6$&F66#FYFZ -F"6$-%'subsopG6%/FY%%NULLG/FOF]uF6FZ-Fat6$-%)dev/sinhGFdt-FbuFht-%&ERRORG6#%%F AILGF0F0F0F0 M6R0 I)dev/tanhR6$%"uG%"nG6*%%factG%"iG%&ifactG%&exponG%(exampleG%%initG%&interG%$si gG6#%RCopyright~Bruno~Salvy,~INRIA~Rocquencourt,~FranceG6"C&@$-%%typeG6$9$%*und efinedG-%'RETURNG6#F9@$4-F66$F8%%listG-F;6#-%%tanhG6#F8>8+-%&evalrG6#-%'SignumG 6#&F86#""$@)/FH"""C(>8$-%(dev/powG6%F8""#9%>8(7*FT""!#!""FQFT#Fen"#:Fen#!#<"$:$ FQ>8&"&?.%>8'"$c#?(8%""%FTFfn%%trueGC%>Fco,$*(FcoFT,&FioFenFTFTFT,&FioFTFTFTFTF en>Ffo,$FfoFjo>Fhn7%-%#opG6#Fhn**FfoFT,&FfoFTF\oFTFT-%*bernoulliG6#,&FioFenFenF TFTFcoF\oFio-F;6#-%(dev/prdG6$F8-%-dev/endofdevG6%FW,&FfnFTFTFTFhn/FHF\oC$>FH-F J6#-FM6#-%+dev/lcoeffGFF@%FS-F;6#-Feq6%-%(dev/expG6$-Fbq6$F8!"#FfnFfn7%FTFjn-%$ seqG6$-Fgp6#7&F[s,&FioFenF\oFTFen,$FioFen/Fio;FT,&-%%iquoG6$FfnFenFTFTFT-F;6#-F eq6%-Fgr6$-Fbq6$F8FenFfnFfn7%F\oFjn-F^s6$-Fgp6#7&FenFcsF[sFdsFes/FHFjnC%>8)-F"6 $&F86#FenFfn>8*-F"6$-%'subsopG6%/Fen%%NULLG/FQFiuF8Ffn-F;6#-Fbq6$-%(dev/addG6$F buF\u-FY6%-F`v6$FT-FbqFavF\oFfn-%&ERRORG6#%%FAILGF2F2F2F2 M6R0 I)dev/cothR6$%"uG%"nG6)%%factG%"iG%(exampleG%&ifactG%%initG%&interG%$sigG6#%RCo pyright~Bruno~Salvy,~INRIA~Rocquencourt,~FranceG6"C&@$-%%typeG6$9$%*undefinedG- %'RETURNG6#F8@$4-F56$F7%%listGC$>8(-%*traperrorG6#-%%cothG6#&F76#"""@%/FC%*last errorGF9-F:6#FC>8*-%&evalrG6#-%'SignumG6#&F76#""$@)/FSFLC'>8$-%(dev/powG6%F7""# 9%>8&7(#FLFfn""!#!"""#XFL#F_o"$X*F_o>8'#""%Fho?(8%FfnFL,&F`oFLFgoFL%%trueGC$>F\ p,$*(F\pFL,&F`pF_oFLFLFgo,&F`pFLFLFLFgoF_o>Fbo7%-%#opG6#Fbo*&F\pFL-%*bernoulliG 6#,&F`pF_oF_oFLFLF`p-F:6#-%(dev/addG6$-F]o6%F7FgoF`o-%(dev/prdG6$F7-%-dev/endof devG6%F[oFapFbo/FSFgoC$>FS-FU6#-FX6#-%+dev/lcoeffG6#F7@%Fhn-F:6#-F^r6%-%(dev/ex pG6$-%/dev/multbyrealG6$F7FgoF`oF`o7%FLFeo-%$seqG6$-F\q6#7$F_oF`p/F`p;FLF`o-F:6 #-F^r6%-F`s6$F7F`oF`o7%FgoFeo-Fgs6$-F\q6#7$!"#F`pF\t/FSFeoC%>FC-F"6$&F76#F_oF`o >8)-F"6$-%'subsopG6%/F_o%%NULLG/FfnFjuF7F`o-F:6#-F[r6$-Ffq6$FL-F[r6$FCFcu-F]o6% -FfqFcvFgoF`o-%&ERRORG6#%%FAILGF1F1F1F1 M6R0 I)dev/sechR6$%"uG%"nG6(%%factG%"iG%(exampleG%&ifactG%%initG%$sigG6#%RCopyright~ Bruno~Salvy,~INRIA~Rocquencourt,~FranceG6"C&@$-%%typeG6$9$%*undefinedG-%'RETURN G6#F7@$4-F46$F6%%listG-F96#-%%sechG6#F6>8)-%&evalrG6#-%'SignumG6#&F66#""$@)/FF" ""C'>8$-%(dev/powG6%F6""#9%>8&7*FR""!#!""FYFR#""&"#CFY#!#h"$?(FO>8'F`o?(8%""%FR FZ%%trueGC$>Fbo,$*(FboFRFdoFR,&FdoFYFjnFRFRFY>Ffn7%-%#opG6#Ffn*&-%&eulerG6#,$Fd oFYFRFboFjnFdo-F96#-%-dev/endofdevG6%FUFZFfn/FFFjnC$>FF-FH6#-FK6#-%+dev/lcoeffG FD@%FQC$>8(-%(dev/expG6$-%/dev/multbyrealG6$F6FjnFZ@%32FZ-%+dev/lengthG6#Fgq0&F gq6#-%%nopsGFcr%)infinityG-F96#-F\r6$FgqFY-F96#-F\r6$-%(dev/prdG6$Fgq-Fip6%-FW6 %FgqFYFZFZ-%$seqG6$-F_p6#7$FRFdo/Fdo;FhnFZFYC$>Fgq-Fiq6$F6FZ@%F_r-F96#-F\r6$Fgq !"#-F96#-F\r6$FbsFjt/FFFhn-FW6%-%)dev/coshGFdtFjnFZ-%&ERRORG6#%%FAILGF0F0F0F0 M6R0 I)dev/cschR6$%"uG%"nG6(%"iG%(exampleG%&exponG%&ifactG%%initG%$sigG6#%RCopyright ~Bruno~Salvy,~INRIA~Rocquencourt,~FranceG6"C&@$-%%typeG6$9$%*undefinedG-%'RETUR NG6#F7@$4-F46$F6%%listGC$>8(-%*traperrorG6#-%%cschG6#F6@%/FB%*lasterrorGF8-F96# FB>8)-%&evalrG6#-%'SignumG6#&F66#""$@)/FO"""C'>8%7(#!""""'""!#""("$g$Fen#!#J"&? ^"""#>8'"$?(>8&"#K?(8$FXFen,&9%FenF[oFen%%trueGC%>Ffo,$*(FfoFen,&F\pFenFenFenFe n,&F\pFdoFenFenFenFdo>Fio,$Fio""%>Fhn7%-%#opG6#Fhn,$*(,&FioFenF[oFenFenFfoF[o-% *bernoulliG6#,&F\pFdoFdoFenFenFdoF\p-F96#-%(dev/addG6$-%(dev/powG6%F6F[oF^p-%(d ev/prdG6$F6-%-dev/endofdevG6%-F[r6%F6FdoF]pF]pFhn/FOF[oC$>FO-FQ6#-FT6#-%+dev/lc oeffGFH@%FZC$>FB-%(dev/expG6$-%/dev/multbyrealG6$F6F[oF^p@%32F^p-%+dev/lengthGF M0&FB6#-%%nopsGFM%)infinityG-F96#-Fes6$FBFdo-F96#-Fes6$-F^r6$FB-Far6%-F[r6%FBFd oF^pF^p7#-%$seqG6$-F\q6#7$FenF\p/F\p;F]oF^pFdoC$>FB-Fbs6$F6F^p@%Fhs-F96#-Fes6$F B!"#-F96#-Fes6$FjtFbv/FOF]o-F96#-F[r6%-%)dev/sinhGF\vF[oF^p-%&ERRORG6#%%FAILGF0 F0F0F0 M6R0 I,dev/arcsinhR6$%"uG%"nG60%%factG%"iG%%sideG%(exampleG%"xG%&ifactG%"jG%"kG%%coe fG%%initG%'newresG%$resG%'tomultG%$sigG6#%RCopyright~Bruno~Salvy,~INRIA~Rocquen court,~FranceG6"C&@$-%%typeG6$9$%*undefinedG-%'RETURNG6#F?@$4-F<6$F>%%listG-FA6 #-%(arcsinhG6#F>>81-%&evalrG6#-%'SignumG6#&F>6#""$@)/FN"""C&>8'7*FZ""!#!""""'FZ #FW"#S""##!"&"$7"FW>8)#Fao"#;?(8%""%FZ9%%%trueGC$>Fdo,$*(FdoFZ,&FhoF_oF[oFZFZFh oF[o#F[oF_o>Fgn7%-%#opG6#Fgn*&FdoFZ,&FhoF_oFZFZF[oFho-FA6#-%(dev/prdG6$F>-%-dev /endofdevG6%-%(dev/powG6%F>F_oFjoFjoFgn/FNF[oC*>8,-%&coeffG6%-%+dev/lcoeffGFL^# FZFin@%-F<6$Fgq%)constantGC$>FN-FP6#-FS6#Fgq@%5FdqFY>8&FN-%&ERRORG6#%%FAILG>F[s FZ>8$-Fbq6%F>!"#Fjo>8/Fin>80FZ>8.-%/dev/multbyrealG6$-%'dev/lnG6$-F]t6$F>,$F[sF _oFjoF[s?(FhoFZFZFjo0FgsF[tC%>FgsF[t>Fis-F\q6$FisFbs>F[t-%(dev/addG6$Fgs-F]t6$F is,$**)F[o,&FhoFZFZFZFZ-%(productG6$,&.8+F_oF[oFZ/Fju;FZFhoFZ-Fgu6$,$FjuF_oF\vF [oFhoF[o#FZF_o-FA6#F[t/FNFinC$>8--F"6$&F>6#F_oFjo@%32Fjo-%+dev/lengthG6#Fgv0&Fg v6#-%%nopsGFaw%)infinityG-FAFawC(>Fgn7$FgvFin>8*-FK6#8(>F[vFZ?(FhoFZFZFjoF[pC%> F]x-%%diffG6$F]xF`x>F[v*&F[vFZFhoF[o>Fgn7%Fdp*&-%%subsG6$/F`xFjvF]xFZF[vFZFho>F bs-%'subsopG6%/F_o%%NULLG/FWFfyF>-FA6#-F_q6%FbsFjoFgnF\sF8F8F8F8 M6R0 I,dev/arccoshR6$%"uG%"nG6.%%factG%"iG%(exampleG%"xG%"jG%"kG%%coefG%%initG%'newr esG%$resG%'tomultG%$sigG6#%RCopyright~Bruno~Salvy,~INRIA~Rocquencourt,~FranceG6 "C&@$-%%typeG6$9$%*undefinedG-%'RETURNG6#F=@$4-F:6$F<%%listGC$>8+-%*traperrorG6 #-%(arccoshG6#F<@%/FH%*lasterrorGF>-F?6#FH>8/-%&evalrG6#-%'SignumG6#&F<6#""$@)/ FU"""F>/FU!""C*>8*-%&coeffG6%-%+dev/lcoeffGFN^#F[o""!@%-F:6$F`o%)constantGC$>FU -FW6#-FZ6#F`o@&F\oF>/FU%%FAILG-%&ERRORG6#FdpF6>8$-%(dev/powG6%F8-Fgo>8.F [o>8,-%'dev/lnG6$-%/dev/multbyrealG6$F<""#F^q?(8%F[oF[oF^q0F`qFdqC%>F`qFdq>Fbq- %(dev/prdG6$FbqFip>Fdq-%(dev/addG6$F`q-Fiq6$Fbq,$*(-%(productG6$,&.8)F[rF]oF[o/ Fas;F[oF]rF[o-F^s6$,$FasF[rFcsF]oF]rF]o#F]oF[r-F?6#Fdq/FUFgoC$>FH-F"6$&F<6#F[rF ^q@%32F^q-%+dev/lengthGFS0&FH6#-%%nopsGFS%)infinityGFRC(>8&7$FHFgo>8(-FM6#8'>Fb sF[o?(F]rF[oF[oF^q%%trueGC%>Fbu-%%diffG6$FbuFeu>Fbs*&FbsF[oF]rF]o>F_u7%-%#opG6# F_u*&-%%subsG6$/FeuF`tFbuF[oFbsF[oF]r>Fip-%'subsopG6%/F[r%%NULLG/FhnF_wF<-F?6#- %-dev/endofdevG6%FipF^qF_uFepF6F6F6F6 M6R0 I,dev/arctanhR6$%"uG%"nG6*%%factG%"iG%(exampleG%"xG%"jG%"kG%%initG%$sigG6#%RCop yright~Bruno~Salvy,~INRIA~Rocquencourt,~FranceG6"C&@$-%%typeG6$9$%*undefinedG-% 'RETURNG6#F9@$4-F66$F8%%listGC$>8*-%*traperrorG6#-%(arctanhG6#F8@%/FD%*lasterro rGF:-F;6#FD>8+-%&evalrG6#-%'SignumG6#&F86#""$@)/FQ"""-F;6#-%(dev/prdG6$F8-%-dev /endofdevG6%-%(dev/powG6%F8""#9%Fdo7#-%$seqG6$-%#opG6#7$*$,&8%FcoFgnFgn!""F_p/F _p;""!Fdo/FQF`pF:/FQFcpC$>FD-F"6$&F86#FcoFdo@%32Fdo-%+dev/lengthGFO0&FD6#-%%nop sGFO%)infinityGFNC(>8&7$FDFcp>8(-FI6#8'>8)Fgn?(F_pFgnFgnFdo%%trueGC%>F\r-%%diff G6$F\rF_r>Far*&FarFgnF_pF`p>Fiq7%-Fjo6#Fiq*&-%%subsG6$/F_rFjpF\rFgnFarFgnF_p>8$ -%'subsopG6%/Fco%%NULLG/FZFjsF8-F;6#-F^o6%FesFdoFiq-%&ERRORG6#%%FAILGF2F2F2F2 M6R0 I,dev/arccothR6$%"uG%"nG6+%%factG%"iG%"xG%(exampleG%"jG%"kG%%initG%%invuG%$sigG 6#%RCopyright~Bruno~Salvy,~INRIA~Rocquencourt,~FranceG6"C&@$-%%typeG6$9$%*undef inedG-%'RETURNG6#F:@$4-F76$F9%%listGC$>8*-%*traperrorG6#-%(arccothG6#F9@%/FE%*l asterrorGF;-F<6#FE>8,-%&evalrG6#-%'SignumG6#&F96#""$@)/FR!""C$>8+-%(dev/powG6%F 9Fhn9%-F<6#-%(dev/prdG6$F[o-%-dev/endofdevG6%-F]o6%F[o""#F_oF_o7#-%$seqG6$-%#op G6#7$*$,&8%Fjo"""FfpFhnFep/Fep;""!F_o/FRFfpF;/FRFipC$>FE-F"6$&F96#FjoF_o@%32F_o -%+dev/lengthGFP0&FE6#-%%nopsGFP%)infinityGFOC(>8'7$FEFip>8(-FJ6#8&>8)Ffp?(FepF fpFfpF_o%%trueGC%>Fbr-%%diffG6$FbrFer>Fgr*&FgrFfpFepFhn>F_r7%-F`p6#F_r*&-%%subs G6$/FerF`qFbrFfpFgrFfpFep>8$-%'subsopG6%/Fjo%%NULLG/FenF`tF9-F<6#-Ffo6%F[tF_oF_ r-%&ERRORG6#%%FAILGF3F3F3F3 M6R0 I,dev/arcsechR6$%"uG%"nG6"6#%RCopyright~Bruno~Salvy,~INRIA~Rocquencourt,~France GF'-%,dev/arccoshG6$-%(dev/powG6%9$!""9%F2F'F'F'F' M6R0 I,dev/arccschR6$%"uG%"nG6"6#%RCopyright~Bruno~Salvy,~INRIA~Rocquencourt,~France GF'-%,dev/arcsinhG6$-%(dev/powG6%9$!""9%F2F'F'F'F' M6R0 I-evalc/RootOfR6#%"pG6,%"xG%"yG%#p1G%#p2G%"aG%"bG%"vG%"cG%$rofG%"gG6#%JCopyrigh t~1993~by~Waterloo~Maple~SoftwareG6"@'5/9#"""33/F7""#-%%typeG6$&9"6#F<-%"=G6$-% *identicalG6#%&labelG%)anythingG-F>6$9$.-%(polynomG6$%'algnumG%#_ZG@%3-F>6$FM.- FP6$%(numericGFS/-%&sturmG6&-%)sturmseqG6$FMFSFS,$%)infinityG!""F^o-%'degreeGF\ o>-%/evalc/rememberG6#-.%'RootOfG6#FM-%,evalc/splitG6$Ffo""!C2>8*-%,evalc/evalc G6#-%%subsG6$/FS,&8$F8*&^#F8F88%F8F8FM>8(-%#opG6$F8F`p>8)-F`q6$F8&-Fep6$/F \qFS-%*resultantG6%F^qFcqFip>8'-Fep6$/FipFS-F\r6%F^qFcqF\q>8--%$gcdG6$Fgq-%%dif fG6$FgqFS@$0FfrF8>Fgq-%$quoG6%FgqFfrFS>Ffr-Fhr6$F_r-F[s6$F_rFS@$F^s>F_r-Fas6%F_ rFfrFS@$/-Fep6$/FSF]pFgqF]p>Fgq-Fas6%FgqFSFS@$/-Fep6$F`tF_rF]p>F_r-Fas6%F_rFSFS >Fgq-Fho6#Fgq>F_r-Fho6#F_r>-FdoF]u-F[p6$FgqF]p>-FdoF`u-F[p6$F_rF]p>Fco-F[p6$F_r FgqF;C$>8+-%*traperrorG6#-Fbp6#F@@'30F^v%*lasterrorG/-F`q6$F8,-FbpFeo@%/F^vFgvFaw-F[p6$-Fgo6$-F`q6#-F`q6$F8Faw-F`q6$F8F^v-Fgo6$ -F`q6#-F`q6$F8$-%,eval c/evalcG6#&F76#F0@%-%%typeG6$F<.-%.evalc/unsplitG6#%*algebraicG-FH6#-.%)Lambert WG6#-%#opG6#F8&-FR6$F1F<>8'-FR6$F4F<@)-FD6$,&FVF1*&^#F1F1FZF1F1.-%(complexG 6#%&floatG-F>6#-FO6#Fjn0FZF9-FH6#-FO6#-%,evalc/stdcxGFS-%'memberG6$-%'signumG6# ,&FVF1*$-%$expGF8!""F1<$F9F1-%,evalc/splitG6$-FO6#FVF9-FH6#F\qF3C%>8%F6>F<-F>6# &F76#F4@%54-FD6$Fbq.%(integerGFC-FH6#-FN6$FbqF[pC&>FVFW>FZFen@$3/FZF92FbqF9C$>8 (-FapF]q@&/F\sF1-%'RETURNG6#-%0evalc/conjugateG6#-F"6$,$FbqFgpFV532FbqFgp/F\sFg p33/FbqFgpF\t/F`pFgp-Fas6#-Fds6#-F"6$,&FbqFgpFgpF1FV@)Fhn-F>6#-FO6$FbqFjnFfo-FH 6#-FOFbr33F_t/F]sFgpF]p-Fjp6$-FO6$FgpFVF9-FH6#-FO6$FbqFVY6$QCexpecting~1~or~2~a rguments,~got~%1F$F0F$F$F$6" M6R0 I(evalc/LR6#%%exprG6"F&F&-%&evalcG6#-%#lnG6#*$,&"""F/9$!""F1F&F&F&F& M6R0 I(evalc/QR6#%%exprG6"F&F&-%&evalcG6#*$,&"""F,9$!""F.F&F&F&F& M6R0 I0evalr/intersectR6#%&listeG6$%"iG%$tmpG6#%aoCopyright~(c)~1990~by~the~Universi ty~of~Waterloo.~All~rights~reserved.G6"@%1""#-%%nopsG6#9$C$>8%-%#opG6$"""F2?(8$ F.F9F/%%trueG@%5-%'memberG6$-%&evalrG6#-%'SignumG6#,&-F76$7$F;F9F2F9-F76$F.F5!" "<$""!F9-F@6$-FC6#-FF6#,&-F76$7$F;F.F2F9-F76$F9F5FN<$FNFP-%'RETURNG6#%%NULLG>F5 ;-FC6#-%$maxG6$FIFen-FC6#-%$minG6$FXFLF2F+F+F+6" M6R0 I(evalr/QR6$%#rgG%&exactG6"F'F'-%*evalr/powG6%-%*evalr/addG6$7#;"""F0-%*evalr/p rdG6%7#;!""F69$9%F6F8F'F'F'F' M6R0 I*evalr/prdR6%%$rg1G%$rg2G%&exactG6%%&isposG%$varG%#t1G6#%aoCopyright~(c)~1990~ by~the~University~of~Waterloo.~All~rights~reserved.G6"@%3-%%typeG6$9$%&rangeG-F 26$9%F5C$@&3/-%#opG6$"""F4-F>6$""#F4-F26$F=%)constantGC%>8&-%&evalrG6#-%'Signum G6#F=@(5/FIF@/FI""!>FI;*&F=F@-F>6$F@F8F@*&F=F@-F>6$FCF8F@/FI!"">FI;FZFW/FI%%FAI LG-%'RETURNG6#;,$%)infinityGFhnFbo@'-%$hasG6$FIFbo-F^o6#-%2simplify/infinityG6# FI53-%(hastypeG6$-F>6$F@FI%&floatG4-F2F`p3-F_p6$-F>6$FCFIFcp4-F2Fhp-F^o6#-%&eva lfGF[p-F^oF[p3/FXFen-F26$FXFFC%>FI-FK6#-FN6#FX@(FQ>FI;FW*&FAF@FXF@Fgn>FI;F_rFWF [oF]o@%FdoFgoFaq@%333FD-F26$FAFFFdq-F26$FenFFC%@$5-%'memberG6$;FTFbo<$F4F8-F_s6 $;FaoFTFbsC$>FI-F"6$-F>6#-%%subsG6%/Fas;F@Fbo/Fes;FaoFhn7$F4F89&@$5/FI;FboFbo/F I;FaoFaoFaq>FI;-FK6#-%$minG6&FWFZF_r*&FAF@FenF@-FK6#-%$maxGFau@%FdoFioFIC(>8%-% $mapG6$R6#%"xGF.F.F.@$/FAF`oF4F.F.F.7#-F>6#-%&unionG6$<#-F>6#-%*evalr/varGFO<#- F>6#-F]w6#FA@$2FT-%%nopsG6#FjuC$>Fju-F>6$F@Fju-F^o6$-F"6$-F>6#-F]t6$/Fju-.%)INT ERVALG6$FjwFesFctFdt-F"6$-F>6#-F]t6$/Fju-Ffx6$FjwFasFctFdt>Fju-F\v6$RF_vF.F.F.@ $FbvF4F.F.F.7#-F>6#-Fgv6$<#-F>6#-F]wF[r<#-F>6#-F]w6#Fen@$FdwC$>FjuFjwF\x>8$-%,e valr/isposG6&,&FZF@F_rFhn%&falseGF^[lFdt@)5/Fiz%%trueG/FizFT-F^o6#-%/evalr/prd/ caseG6%F4F8Fdt/FizF^[l-F^o6#-Fg[l6%F8F4Fdt/FizF\oF]o-F^o6$-F\v6%R6$F`v%"yGF.F.F .-Fg[l6$-F>6#F4F8F.F.F.-%,evalr/chvarG6$Fct-F>6$F@FizFdt-F\v6%RFd\lF.F.F.Ff\lF. F.F.-F[]l6$7$F8F4-F>6$FCFizFdtC$>FI-F\v6&R6%F`vFe\l%"zGF.F.F.-F>6#-F\v6&F"F8F4F dtF.F.F.F4F8Fdt@%/-FfwF[pF@FI-%,evalr/unionGF[pF.F.F.6", I/evalr/prd/caseRF$6#F)F,F.C$>Fiz-F[[l6&,&FWF@FbuFhnF^[lF^[lFdt@)F`[l-F^o6#-%0e valr/prd/case1GFh[lFi[l-F^o6#-%0evalr/prd/case2GFh[lF^\lF]o-F^o6$-F\v6%RFd\lF.F .F.-Fd_lFg\lF.F.F.Fj\lFdt-F\v6%RFd\lF.F.F.-Fh_lFg\lF.F.F.-F[]l6$FctFe]lFdtF.F.F .6", I0evalr/prd/case1RF$Fj^lF,F.C$>Fiz-F[[l6&,&FWF@FZFhnF^[lF^[lFdt@)F`[l-F^o6#-%1e valr/prd/case11GFh[lFi[l-F^o6#-%1evalr/prd/case12GFh[lF^\lF]o-F^o6$-F\v6%RFd\lF .F.F.-FaalFg\lF.F.F.Fj\lFdt-F\v6%RFd\lF.F.F.-FealFg\lF.F.F.Fc`lFdtF.F.F.6", I0evalr/prd/case2RF$Fj^lF,F.C$>Fiz-F[[l6&,&FZF@FbuFhnF^[lF^[lFdt@)F`[l-F^o6#-%1 evalr/prd/case21GFh[lFi[l-F^o6#-%1evalr/prd/case22GFh[lF^\lF]o-F^o6$-F\v6%RFd\l F.F.F.-F\clFg\lF.F.F.Fj\lFdt-F\v6%RFd\lF.F.F.-F`clFg\lF.F.F.Fc`lFdtF.F.F.6", I1evalr/prd/case11RF$Fj^lF,F.C$>Fiz-F[[l6&,&F_rF@FbuFhnF^[lF^[lFdt@)F`[l-F^o6#; FbuFWFi[l-F^o6#FarF^\l-F^o6#F\o-F^o6$-F>6#-F[]l6$FfdlF]]l-F>6#-F[]l6$FarFe]lF.F .F.6", I1evalr/prd/case12RF$Fj^lF,F.C$>FizF`dl@)F`[l-F^o6#;FbuFZFi[l-F^o6#;F_rFZF^\lF] o-F^o6$-F>6#-F[]l6$F]flF]]l-F>6#-F[]l6$F`flFe]lF.F.F.6", I1evalr/prd/case21RF$Fj^lF,F.C$>Fiz-F[[l6&,&F_rF@FWFhnF^[lF^[lFdt@)F`[l-F^o6#FV Fi[lF^flF^\lF]o-F^o6$-F>6#-F[]l6$FVF]]l-F>6#-F[]l6$;*&F=F@FAF@FZFe]lF.F.F.6", I1evalr/prd/case22RF$Fj^lF,F.C$>FizF`gl@)F`[l-F^o6#;FWFbuFi[l-F^o6#;F_rFbuF^\lF ]o-F^o6$-F>6#-F[]l6$FjhlF]]l-F>6#-F[]l6$;FahlFbuFe]lF.F.F.6" M6R0 I(evalr/LR6$%#rgG%&exactG6"F'F'-%)evalr/lnG6$-%*evalr/powG6%-%*evalr/addG6$7#;" ""F3-%*evalr/prdG6%7#;!""F99$9%F9F;F;F'F'F'F' M6R0 I*evalr/addR6$%$rg1G%$rg2G6#%#t1G6#%aoCopyright~(c)~1990~by~the~University~of~W aterloo.~All~rights~reserved.G6"@'3-%%typeG6$9$%&rangeG-F/6$9%F2C$>8$;,&-%#opG6 $"""F1F>-F<6$F>F5F>,&-F<6$""#F1F>-F<6$FDF5F>@%-%$hasG6$F8%)infinityG-%2simplify /infinityG6#F8F85/F1%%FAILG/F5FQFQ-%,evalr/unionG6#-%$mapG6%R6$%"xG%"yGF+F+F+-F <6#-FW6%F"F5F1F+F+F+F1F5F+F+F+6" M6R0 I/evalr/signuminR6"6%%"iG%%lexpG%"rG6#%aoCopyright~(c)~1990~by~the~University~o f~Waterloo.~All~rights~reserved.GF$C'@$/9#"""-%'RETURNG6#9"@%-%$hasG6$7#F3%)inf inityG@'-%'memberG6$,$F9!""F8-F16#F>-F<6%F9F8.8$-F16#-F"6#-%#opG6#-%'subsopG6$/ FE%%NULLGF8C$>8&-%2simplify/infinityG6#-%$minGF2@%3-%%typeG6$FT.%)functionG/-FK 6$""!FT.FY>8%7#-FK6#FT>Fao7#FT>FaoF8>FT<#&Fao6#F/?(FE""#F/-%%nopsG6#Fao%%trueG> FT-%$mapG6%R6$%"xG%"yG6#%"sGF$F$C$>FE-%&evalrG6#-%'SignumG6#,&9%F/9$F?@'5/FEF// FEF^oFfq/FEF?Feq6$FfqFeqF$F$F$FT&Fao6#FE-FY6#FcoF$F$F$6", I/evalr/signumaxRF$F%F)F$C'@$F-F0@%F5@'-F<6$F9F8-F16#F9-F<6%F>F8FD-F16#-FbrFIC$ >FT-FV6#-%$maxGF2@%3Ffn/F\o.Ffs>FaoFbo>FaoFfo>FaoF8>FTFio?(FEF]pF/F^pFap>FT-Fdp 6%RFgpFjpF$F$C$>FEF^q@'5F[rFjqFfqFiqFeqF\rF$F$F$FTF]r-FfsF`rF$F$F$6", I0evalr/signumabsR6#%%exprG6#F(F)F$C$>FE-F_q6#-Fbq6#Ffq@)FiqFfqFjqF^oF[r,$FfqF? -%$absGFduF$F$F$6", I,evalr/evalrR6$F]u%&exactG6&F&%"fGF(%$idxG6$%)rememberGF*F$C$@$-F66$Ffq%%FAILG -F16#Fgv@33/-F_p6#-%*intersectG6$-Fdp6$R6#FhpF$F$F$-FK6$F^oFfqF$F$F$-%'indetsG6 $FfqFjn<&F_oFjs.Fhu.FbqF^o-Fgn6$Ffq.%)realconsG@'-F66$FfqF9C$>FT-FVFdu-.%)INTER VALG6#;FTFTFeq-Fix6#;FfqFfqC$>FT-%&evalfGFduFhx-Fgn6$Ffq%"+GC$>FT-Fju6$-FK6$F/F fqFeq?(FEF]pF/-F_pFduFap>FT-%*evalr/addG6$FT-Fju6$-FK6$FEFfqFeq-Fgn6$Ffq%"*GC$> FTFiy?(FEF]pF/F^zFap>FT-%*evalr/prdG6%FTFczFeq-Fgn6$Ffq%"^G@%-Fgn6$-FK6$F]pFfq. %)rationalG-%*evalr/powG6%FiyFg[lFeq-Fju6$-%$expG6#*&Fg[lF/-%#lnG6#F[zF/Feq3-Fg n6$FfqFin0FfwFixC$>FaoFfw@+5/Fao.%$IntG/Fao.%$intG-%*evalr/intG6$-FKFduFeq334-% (hastypeG6$Ffq.%&rangeG/-F_p6#-%&minusG6$-Fiw6$Ffq.%%nameG-%&unionG6$<#%*consta ntsG<#%#_ZGF^o-F<6$Fao7'F_oFjsF\xF]x.%&shakeG@+/FaoF_o-F"6#Fh]l/FaoFjs-FbrFj_l/ FaoF\x-FjtFj_l/FaoF]x-%-evalr/SignumGFj_l-%,evalr/shakeG6$Fh]l%'DigitsG/FaoFe_l Fgv/Fao.%'RootOfG@%3/F^zF]p-Fgn6$Fg[l.-%(complexG6#%(numericG-Fc`l6$Fg[lFe`lC$F hv-%&ERRORG6#%@Not~implemented,~~~evalr/RootOfGC$>Fao-%$catG6$%'evalr/GFao@)4-F gn6$Fao.%*procedureG-Fhal6$%0Not~implementedGFao5/F^zF/3F\al-F<6$Ffw<&.%$PsiG.% )LambertWG.%%erfcG.%#EiGC$@%F[cl>8'FQ>F[dlF[z-Fao6%F[dl-Fju6$-FK6$F^zFfqFeqFeq- F<6$Ffw<%.%&GAMMAG.%%BetaG.%%ZetaG-Fhal6#%Bmultivariate~case~not~implementedG-F ao6$-FjuFg]lFeq-FgnFh^l-Fix6#;-Fix6$Ffq;F>F9Ffel3-Fgn6$Ffq-%)specfuncG6$%)anyth ingGFix/-Fdp6$RFewF$F$F$-FgnF^^lF$F$F$<#Fh]l<#FapFfq33Fjel-Fgn6$F[zFi^l/-Fdp6$R FewF$F$F$FdflF$F$F$<#-FK6#-FN6$/F/FQFfqFfflC$>FT-Fdp6%FjuFg[lFeq-Fix6#;-Fix6$F[ z;-FK6$F/-FK6#-FK6$F/FT-FK6$F]p-FK6#-FK6$F]pFTF\hl-Fhal6$%-Invalid~typeGFfqF$F$ F$6" M6R0 I,evalr/shakeR6$%%exprG%$digG6#%"sG6#%aoCopyright~(c)~1992~by~the~University~of ~Waterloo.~All~rights~reserved.G6"C&>%'DigitsG,&9%"""""#F1>8$-%*traperrorG6#-%, shake/shakeG6$9$-%&FloatG6$"#5,$F0!""-%'subsopG6$/""%%%NULLG-%#opG6#F9@%/F4%*la sterrorG-%&ERRORG6#F4-%'RETURNGFPF+F+F+6", I,shake/shakeR6$F%%%amplG6)%"rG%"iG%$resG%&expr2G%"fGF(%$idxG6$F*%)rememberGF+@ /5-%%typeG6$F;.%(numericG-%'memberG6$F;<#%*constantsG@'/F;^#F1-FO6#%%FAILG2""!- %&evalfG6#F;-.%)INTERVALG6#;-F`p6#*&F;F1,&F1F1F0FAF1-F`p6#*&F;F1,&F1F1F0F1F1-Fc p6#;F[qFgp-F^o6$F;%"+GC$>F4-F96$-FI6$F1F;F0?(8%F2F1-%%nopsGFap%%trueG>F4-%*eval r/addG6%F4-F96$-FI6$F\rF;F0%&falseG-F^o6$F;%"*GC$>F4Fgq?(F\rF2F1F]rF_r>F4-%*eva lr/prdG6&F4FdrFhrFhr-F^o6$F;%"^G@%-F^o6$-FI6$F2F;.%)rationalGC%>8'Fgq>8&-FcpF+? (F\rF1F1-F^r6#F_tF_rC$@%0-FI6$7$F\rF1F_t-FI6$7$F\rF2F_t>F4-Fcp6#-FI6$F\rF_t>F4- Fcp6#;,&FitF1F0FA,&F\uF1F0F1>Fat-Fcp6$-FI6#Fat-FI6#-%*evalr/powG6%F4FisFhr-F96$ -%$expG6#*&FisF1-%#lnG6#FiqF1F0-F^o6$F;.%)functionG@)/-FI6$F^pF;.%'RootOfG@%3/F ]rF2-F^o6$FisFao-F96$F_pF0Fjo33/Fcw.%$absG-F^o6$-FIFapFew/-F^r6#FexF2@%4-%$hasG 6$-FI6$F2FexFio-F96$-Fbx6#F^y-%$maxG6$-FC6$/F1F1F^yF0-F96$-%%sqrtG6#,&*$-%&coef fG6%F^yFioF^pF2F1*$-Fbz6%F^yFioF1F2F1-Fey6%-FC6$FiyFaz-FC6$FiyFezF033F`x-%(hast ypeGFdx-F`[l6$Fex.%&floatG-F96$-%%evalG6#-%%subsG6$/FewR6$%"xG%"yGF+F+F+@%/9#F2 F0-FfwFapF+F+F+F;F0C$>8(-%$catG6$%'evalr/GFcw@'4-F^o6$Fh\l.%*procedureG-FO6$%0N ot~implementedGFh\l5/F]rF13Fiw-Fco6$Fcw<&.%$PsiG.%)LambertWG.%%erfcG.%#EiGC&@%F g]l>8*FG>Fg^lFiq>F_t-F96$-FI6$F]rF;F0>FatFbt?(F\rF1F1FdtF_rC&@%Fht>F4F`u>F4Feu> 8)-Fh\l6%Fg^lF4Fhr@$/Fe_lF\pYF\p@%/-F^r6#Fe_lF1>Fat-Fcp6$F]v;-F`p6#-FI6$F1-FIF^ `l-F`p6#-FI6$F2Fg`l>Fat-Fcp6$F]vFg`l-FO6#%Bmultivariate~case~not~implementedG33 3-F^o6$F;-%)specfuncG6$%)anythingGFcpFiw-F^o6$Fiq.%%nameG-F^o6$Fis.%&rangeG-FC6 $/F1FGF;-FO6$%-Invalid~typeGF;F+F+F+6" M6R0 I-evalr/univarR6%%+expressionG%$ex1G%$ex2G6,%"xG%#rgG%%exprG%&derivG%$resG%"iG% $errG%#t1G%#t2G%#t3G6#%aoCopyright~(c)~1991~by~the~University~of~Waterloo.~All~ rights~reserved.G6"C.>8$-%#opG6#-%&minusG6$-%'indetsG6$9$%%nameG<#%*constantsG@ %-%(hastypeG6$FB%&rangeGC%>8%-F@FI>8&-%%subsG6$7#-%$seqG6$/-.%)INTERVALG6$F88)F 8/FgnFMFB@%/-%%nopsG6#FM""">FM-F:F]o>FM;-%$minG6#-FV6$-F:6$F^oFgnFhn-%$maxG6#-F V6$-F:6$""#FgnFhnC$>FM;,$%)infinityG!""Ffp>FPFB>8'-%%diffG6$FPF8>8(7#-%&solveG6 $FjpF8?(FgnF^oF^oF51Fgn-F\o6#F_q@$5/-%&evalrG6#-%'SignumG6#,&-F:6$F^oFMF^o-F:6$ FgnF_qFgpF^o/-F\r6#-F_r6#,&FdrF^o-F:6$FapFMFgpF^oC$>F_q-%'subsopG6$/Fgn%%NULLGF _q>Fgn,&FgnF^oFgpF^o>F_q<#-F:Fgq@%-FH6$Fjp%&floatGC$>8*<#-%*traperrorG6#-%'fsol veG6%FjpF8FM@%55/F`t<#%*lasterrorG-%$hasG6$F`t.Fft1-F\o6#F`tFfq>F_qFhs>F_q<#-F: Fdu>F_qFhs>8+-%$mapG6&R6%F)%"yG%"zGF5F5F5-FR6$/9&FB9%F5F5F5F_qFPF8@%Fgv>8,<#""! >Fjv<#-%&limitG6%FP/F8Fbr%&rightG@%Ffv>8-F[w>Ffw<#-F`w6%FP/F8F\s%%leftG>F[v-F:6 #-%&unionG6$-Fax6$FfwFjvF[v-FZ6#;-F\r6#-Fdo6#F[v-F\r6#-F[pF[yF5F5F56" M6R0 I(evalf/OR6#%"xG6"F&F&-%"OG6#-%&evalfG6#9$F&F&F&F& M6R0 I)series/LR6$%%exprG%"xG6"F'F'-%'seriesG6$-%#lnG6#*$,&"""F09$!""F29%F'F'F'F' M6R0 I)series/QR6$%%exprG%"xG6"F'F'-%'seriesG6$*$,&"""F-9$!""F/9%F'F'F'F' M6R0 I'diff/QR6$%"aG%"xG6"6#%RCopyright~Bruno~Salvy,~INRIA~Rocquencourt,~FranceGF'*& -%"QG6#9$""#-%%diffG6$F.9%"""F'F'F'F' M6R0 I'diff/LR6$%"aG%"xG6"6#%RCopyright~Bruno~Salvy,~INRIA~Rocquencourt,~FranceGF'*& -%%diffG6$9$9%""",&F0F0F.!""F2F'F'F'F' M6R0 I1conjugate/RootOfR6$%"xG%"yG6"F'F'@%3/9#""#-%%typeG6$9%.-%(complexG6#%(numeric G-%'RootOfG6$9$-%*conjugateG6#F0-.F;6#-F76#9"F'F'F'F' M6R0 I%gfun`6Rc.algfuntoalgeq$:-1c.algebraicsubs$:-2c.algeqtodiffeq$:-3c.algeqtoseri es$:-4c&borel$:-5c.cauchyproduct$:-6c.diffeq+diffeq$:-7c.diffeq*diffeq$:-8c,dif feqtable$:-9c2diffeqtohomdiffeq%:-10c,diffeqtorec%:-11c)guesseqn%:-12c(guessgf% :-13c0hadamardproduct%:-14c0holexprtodiffeq%:-15c)invborel%:-16c(Laplace%:-17c, listtoalgeq%:-18c-listtodiffeq%:-19c0listtohypergeom%:-20c+listtolist%:-21c.lis ttoratpoly%:-22c*listtorec%:-23c-listtoseries%:-24c2listtoseriestable%:-25c,max degcoeff%:-26c*maxdegeqn%:-27c,maxordereqn%:-28c,mindegcoeff%:-29c*mindegeqn%:- 30c,minordereqn%:-31c*optionsgf%:-32c&pade2%:-33c,poltodiffeq%:-34c)poltorec%:- 35c/ratpolytocoeff%:-36c(rec+rec%:-37c(rec*rec%:-38c,rectodiffeq%:-39c,rectohom rec%:-40c*rectoproc%:-41c.seriestoalgeq%:-42c/seriestodiffeq%:-43c2seriestohype rgeom%:-44c-seriestolist%:-45c0seriestoratpoly%:-46c,seriestorec%:-47c/seriesto series%:-48b6#%+thismoduleG6gn%3algeqtoseries/doitG%:algeqtoseries/prettyprintG %/algfuntoalgeq2G%7algfuntoalgeq/formpolyG%2algfuntoalgeq/addG%2algfuntoalgeq/m ulG%5algfuntoalgeq/RootOfG%.borelinvborelG%/cheapgausselimG%1diffeqtorec/doitG% -firstnonzeroG%-formatdiffeqG%,formatpoleqG%*formatrecG%,funtodiffeqG%,gensolve linG%(getnameG%6goodinitvalues/diffeqG%3goodinitvalues/recG%.guessandcheckG%+in dicialeqG%(inicondG%,isholonomicG%'lindepG%-listprimpartG%1listtoseries/egfG%5l isttoseries/LaplaceG%1listtoseries/ogfG%4listtoseries/revogfG%4listtoseries/rev egfG%4listtoseries/lgdogfG%4listtoseries/lgdegfG%(l2r/l2rG%(l2h/l2hG%+makediffe qG%(makerecG%)maxindexG%)minindexG%(mygcdexG%)myisolveG%*nbinicondG%+powcompose G%1powcomposesimpleG%5powcomposesimpledoitG%*powdivideG%*powrevertG%,powtruncat eG%/pprimeknowingpG%-rechomrecbisG%1rectodiffeq/doitG%,pade2/exminG%+pade2/doit G%4ratpolytocoeff/elmtG%,ratsolvelinG%/rectohomrecbisG%(s2d/s2dG%(s2a/s2aG%,sys tomatrixG%*typecheckG6#%(packageG6R%.algfuntoalgeqG%.algebraicsubsG%.algeqtodif feqG%.algeqtoseriesG%&borelG%.cauchyproductG%.diffeq+diffeqG%.diffeq*diffeqG%,d iffeqtableG%2diffeqtohomdiffeqG%,diffeqtorecG%)guesseqnG%(guessgfG%0hadamardpro ductG%0holexprtodiffeqG%)invborelG%(LaplaceG%,listtoalgeqG%-listtodiffeqG%0list tohypergeomG%+listtolistG%.listtoratpolyG%*listtorecG%-listtoseriesG%2listtoser iestableG%,maxdegcoeffG%*maxdegeqnG%,maxordereqnG%,mindegcoeffG%*mindegeqnG%,mi nordereqnG%*optionsgfG%&pade2G%,poltodiffeqG%)poltorecG%/ratpolytocoeffG%(rec+r ecG%(rec*recG%,rectodiffeqG%,rectohomrecG%*rectoprocG%.seriestoalgeqG%/seriesto diffeqG%2seriestohypergeomG%-seriestolistG%0seriestoratpolyG%,seriestorecG%/ser iestoseriesG6"Fau6'%/type/gfun/freeG%6type/gfun/has2diffeqsG%3type/gfun/identit yG%1type/gfun/initeqG%-gfun/versionGFauFau6gnc3algeqtoseries/doit%:-49c:algeqto series/prettyprint%:-50c/algfuntoalgeq2%:-51c7algfuntoalgeq/formpoly%:-52c2algf untoalgeq/add%:-53c2algfuntoalgeq/mul%:-54c5algfuntoalgeq/RootOf%:-55c.borelinv borel%:-56c/cheapgausselim%:-57c1diffeqtorec/doit%:-58c-firstnonzero%:-59c-form atdiffeq%:-60c,formatpoleq%:-61c*formatrec%:-62c,funtodiffeq%:-63c,gensolvelin% :-64c(getname%:-65c6goodinitvalues/diffeq%:-66c3goodinitvalues/rec%:-67c.guessa ndcheck%:-68c+indicialeq%:-69c(inicond%:-70c,isholonomic%:-71c'lindep%:-72c-lis tprimpart%:-73c1listtoseries/egf%:-74c5listtoseries/Laplace%:-75c1listtoseries/ ogf%:-76c4listtoseries/revogf%:-77c4listtoseries/revegf%:-78c4listtoseries/lgdo gf%:-79c4listtoseries/lgdegf%:-80c(l2r/l2r%:-81c(l2h/l2h%:-82c+makediffeq%:-83c (makerec%:-84c)maxindex%:-85c)minindex%:-86c(mygcdex%:-87c)myisolve%:-88c*nbini cond%:-89c+powcompose%:-90c1powcomposesimple%:-91c5powcomposesimpledoit%:-92c*p owdivide%:-93c*powrevert%:-94c,powtruncate%:-95c/pprimeknowingp%:-96c-rechomrec bis%:-97c1rectodiffeq/doit%:-98c,pade2/exmin%:-99c+pade2/doit&:-100c4ratpolytoc oeff/elmt&:-101c,ratsolvelin&:-102c/rectohomrecbis&:-103c(s2d/s2d&:-104c(s2a/s2 a&:-105c,systomatrix&:-106c*typecheck&:-107Fau M6R0 I/type/gfun/freeR6$%"xG%"yG6"F'F'4-%$hasG6$9$9%F'F'F'F' M6R0 I6type/gfun/has2diffeqsR6#%"xG6"F&F&-%'memberG6$9$<%.%%erfcG.%'AiryAiG.%'AiryBi GF&F&F&F& M6R0 I3type/gfun/identityR6#%"xG6"F&F&3-%%typeG6$9$%"=G/-%#opG6$"""F+-F/6$""#F+F&F&F &F& M6R0 I1type/gfun/initeqR6$%%exprG%"yG6#%"fG6"F)@%4-%%typeG6$9$-%)functionG6#""!%&fal seGC$>8$-%#opG6$F3F/55/F79%/F7.-%"DG6#F>3-F-6$F7-F16#-%*identicalGFC-F-6$-F96$F 3F7-%#@@G6$-FJ6#FB%(integerGF)F)F)F) M6R0 I-gfun/version$"$m#!"#6" M6R0 I%:-37R6%%$eq1G%$eq2G%%yofzG6&%#y1G%#y2G%"yG%"zG6"F-C$-T#6%9&8&8'-T%6&,&-8$6#F4 """-8%F;F<7$-%%subsG6$/F3F:9$-FA6$/F3F>9%7$F9F=F2F-F-6&%(getnameGc(getname%:-65 %,poltodiffeqGc)poltorec%:-356#/%+modulenameG%%gfunG M6R0 I%:-70R6&%"sG%$eqnG%"yG%"xG6%%&orderG%$deqG%"iG6"F-C$>8%-%'selectG6%%$hasG-%'in detsG6$9%.-%%diffG6$%)anythingG-%*identicalG6#9'-9&F@@%/F0<"F8C$?(8$"""FJF-0F0< #FB>F0-%%subsG6$/-F;6$FBFAFBF0<$-%$seqG6$/---%#@@G6$%"DG8&6#FC6#""!*&-%&coeffG6 %9$FAFinFJ-%*factorialG6#FinFJ/Fin;F\o,&FIFJ!"#FJF8F-F-F-F- M6R0 I%:-22R6"6-%'resultG%#exG%(methodsG%'methodG%"sG%"xG%%bigOG%$ordG%$nbzG%+tryfac torsG%(nbzerosGF$F$C(@%/&9"6#""".%(stampedGC%>8%&F56#""#>8)&F56#""$>8&&F56#""%- %'RETURNG6#-T%6#-T#6$FDF5>8--%%typeG6$F<-%%listG6#%)rationalG>8*-%%nopsG6#F<>8. ,&FhnF7-Fin6#-%%subsG6$/""!%%NULLGF8(-%*traperrorG6#-T'6&F8 F8+-%&orderG6#F\p>8,-%$maxG6$-%#opG6$F?F\pF\oC$>Fcq,&FgnF7FfoF7>Fhq F\o>8$-T)6(7$F7F\pFAFcq,(FcqF7FhqFfoFfoF7FT,&FcqF7FDF7@$0Fdr%%FAILGC$-Fhp6(F?Fj p.%$TheGFho.%,seems~to~beGFdr-FK6#7$,$*&&FdrF6F7&FdrF>FfoFfoFhoF]sF$F$6*%*typec heckGc*typecheck&:-107%.listtoratpolyGc.listtoratpoly%:-22%-listtoseriesGc-list toseries%:-24%.guessandcheckGc.guessandcheck%:-686#/%+modulenameGF[q M6R0 I$:-6R6%%%rec1G%%rec2G%%uofnG6+%#y1G%#y2G%"yG%"zG%#d1G%#d2G%%indsG%"jG%"iG6"F2C '>8(-T#6%9$9&-8$6#8'>8)-F76%9%F:-8%F=>8*-%*intersectG6$-%'indetsG6$F5&%#_CG6#%) anythingG-FL6$F@FN@$0FG<"C$>8+-%$maxG6#-%#opG6#-%$mapG6$Fhn-%&unionGFJ>F@-%%sub sG6$7#-%$seqG6$/&FG6#8,&FO6#,&FY"""FjoF^p/Fjo;F^p-%%nopsG6#FGF@-T'6%-T%6&*&F;F^ pFDF^p7$F5F@7$F;FD-8&F=F]qF:F2F26(%,rectodiffeqGc,rectodiffeq%:-39%,poltodiffeq Gc,poltodiffeq%:-34%,diffeqtorecGc,diffeqtorec%:-116#/%+modulenameG%%gfunG M6R0 I%:-93R6&%"pG%"qG%"xG%"nG6&%"uG%"bG%"iG%"jG6"F.C%@$/-%%subsG6$/9&""!9%F7-%&ERRO RG6#%0not~implementedG?(8&F7"""9'%%trueGC$>&8%6#F>-%&coeffG6%F8F6F>>&8$FF*&,&-F H6%9$F6F>F?-%(convertG6$7#-%$seqG6$*&&FL6#8'F?&FE6#,&F>F?Ffn!""F?/Ffn;F7,&F>F?F jnF?%"+GFjnF?&FE6#F7Fjn-FS6$7#-FW6$*&FKF?)F6F>F?/F>;F7F@F^oF.F.F.F. M6R0 I%:-45R6"6'%"sG%%methG%"lG%"xG%"iGF$F$C(@$0&9"6#""".%(stampedG-%'RETURNG6#-T%6# -T#6$"")F/>8$&F/6#""#>8%&F/6#""$>8'-%#opG6$""!F?@%/-FK6$,&-%%nopsG6#F?F1!""F1F? -%"OGF0>8&7#-%$seqG6$-%&coeffG6%F?FI8(/F\o;FM,&-FK6$FSF?F1FVF1>FZ7#-Fgn6$Fin/F\ o;FMF`o@%/FD.%$ogfGFZ-F86%F2-T'6&F2FZFIFDFjoF$F$6(%*typecheckGc*typecheck&:-107 %-seriestolistGc-seriestolist%:-45%-listtoseriesGc-listtoseries%:-246#/%+module nameG%%gfunG M6R0 I%:-78G6" M6R0 I%:-30""#6#/%+modulenameG%%gfunG M6R0 I%:-63R6$%%exprG%%yofxG66%"xG%"yG%$funG%$deqG%$limG%%deq2G%$ordG%*initcondsG%"i G%(initptsG%%condG%'tabrefG%#nuG%%infoG%#eqG%&pinitG%&limptG%%newxG%%singG%'iss ingG6"F8&-%#opG6$""!9$@$4-%$hasG6$FJFC-%'RETURNG6#,&FA"""FJ!"">8 /FE@$3-%%typeG6$FW%1gfun/has2diffeqsG/-%%nopsG6#FJ""#>FW-%$catG6$FWF\o>80-FG6$; FT,&FinFTFUFTFJ>81-&T%6#FW6%FBFCFbo>8'&Fho6#FT>8(-%&limitG6%-FG6$FinFJ/FCFI%&ri ghtG@$-FN6$Fcp%)infinityG>FcpF^q>87-%'memberG6$Fcp&Fho6#F\o@'/FcpFI>8)F_p0FcpF^ q>Fjq-T'6%F_p,(FBFTFcpFUFCFU-FB6#FC>Fjq-F^r6%F_p,&*&FBFTFCFTFTFUFTFar@%5Faq3Fhn 4-Fen6$Fbo%(integerG>8*FI>F`s,&-Fjn6#-T)6#7$FjqFAFT!"#FT>86-%'selectG6%R6$%"aG% "lGF<6$%)operatorG%&arrowGF<-%&evalbG6#/-FG6$FTFJFAF8 -<#-%$seqG6$8,/Fhu;FI,&F`sFTFUFT>Fcu<#-Ffu6$,&FhuFT-FG6$F\o-FG6#F[tFTFiu@%55/Fc pF^q2FTFin0F[tFauC%@%FhvC$>84FI>85*$FCFUC$>F_wFcp>FawFC@%-Fcq6$FIFcuC$>8.<#/-FB 6#FI-Fep6$-FE6$FboFaw/FCF_w>Fcu-%&minusG6$Fcu<#FI>F[xFau>8+-%&unionG6$F[x<#-Ffu 6$/---%#@@G6$%"DGFhu6#FBF_x-Fep6$-%%diffG6$Fbx-%"$G6$FCFhuFdx/FhuFcu>F\y<#-Ffu6 $/Fdy--Ffy6#FE6#FcpFcz@$4-Fen6$Fjq%$setG>Fjq<#Fjq>Fjq-F^y6$Fjq-%)simplifyG6$F\y F^q@%F[rC$>82-T+6&FgpFA.83.%*algebraicG-F^r6&Fjq-%%subsG6$/FB,&FBFTFcpFTF]\lFar -F^y6$<#/F^xFI-%'removeG6%FNFb\lF^xC$>F]\l-F_\l6%*$FgpFUFAFc\l-F^r6&FjqF]\lFarF ^]lF 8'&F96#""#>8(&F96#FG>8)&F96#""%-%)userinfoG6%F;.%%gfunG%NTrying~to~find~a~ratio nal~generating~functionG?&8,FR%%trueGC&>&8-6#Fgn-%*traperrorG6#-T'6$FIFgn@$/F[o %*lasterrorG\>8&-T)6&FFio-T+F\p@$FapF cp-FW6%F;FY%PTrying~to~find~an~algebraic~generating~functionG?&FgnFRFhnC->Fio-T -6&F807#-%&s olveG6$FfpFiq@$/-%%nopsG6#FbrF;-F?6#7$FbrFgn>8/""!>81Fas>8+-T/6$F[oFN?(8.F;F;Fi rFhnC$>82-%'seriesG6%,&&Fbr6#FjsF;Fes!""FN-Fjr6#F[o@&/F]tFas-F?6#7$FbtFgn3-%%ty peG6$F]tF_t2F`s-%#opG6$FLF]tC$>F`sFau>FcsFjs@$/FcsFasFgoFit-FW6%F;FY%NTrying~to ~find~a~linear~differential~equationG?&FgnFRFhnC'>8$-T1Fhq@$/F_vFbpFgo-FW6%F;FY %3Trying~to~solve~itG>Fio-%'dsolveG6$-Fbu6$F;F_vFiq@$3Fap0Fio%%NULLGC'>8*-%&min usG6$-Few6$-%'indetsG6$-Fbu6$FLFio%%nameG-Fjw6$FIF^x<$%*constantsGFN@$/Fcw<"-F? 6#7$F\xFgn>Fes-F_t6%F\xFN,&-Fjr6#FIF;F;F;>Fes-Fer6$<#-%$seqG6$,&-%&coeffG6%FesF N,&FjsF;FdtF;F;-Fbu6$FjsFIFdt/Fjs;F;F]yFcw@$30FesF`w-F^u6$Fes%$setG-F?6#-%%subs G6$FesFhxFbpF$F$62%*typecheckGc*typecheck&:-107%(guessgfGc(guessgf%:-13%+listto listGc+listtolist%:-21%.listtoratpolyGc.listtoratpoly%:-22%0listtohypergeomGc0l isttohypergeom%:-20%,listtoalgeqGc,listtoalgeq%:-18%-listtoseriesGc-listtoserie s%:-24%-listtodiffeqGc-listtodiffeq%:-196#/%+modulenameGFZ M6R0 I&:-101R6%%"gG%"xG%"nG6&%"kG%"aG%"cG%"iG6"F-@'-%%typeG6$9$%)functionG--%#opG6$" "!F26$-T#6%-F66$"""F29%9&-F66$;""#-%%nopsG6#F2F2-F06$F2-%(polynomG6$%)anythingG F@F8C)>8$-%'selectG6%%$hasG-%'indetsG6$F2%"^GF@@$0-FG6#FQF?-%&ERRORG6&%5report~ this~as~a~bugGF2F@FA>FQ-F6Fgn>8%,&F@F?-F66$F?FQ!"">8&*&F2F?FQFco>FQ,$-F66$FEFQF co*,FeoF?),$F_oFcoFQFco)F_o,$FAFcoF?-%(convertG6$7#-%$seqG6$,&FAF?8'F?/Fhp;F?,& FQF?FcoF?%"*GF?-%*factorialG6#F[qFcoF-F-6$%4ratpolytocoeff/elmtGc4ratpolytocoef f/elmt&:-101F- M6R0 I%:-86R6%%$recG%"uG%"nG6"F(F(,&-%$minG6#-%#opG6#-%$mapG6$F.-%'indetsG6$9$-%)spe cfuncG6$-%'linearG6#9&9%"""F=!""F(F(F(F( M6R0 I%:-53G6" M6R0 I%:-38R6%%$eq1G%$eq2G%%yofzG6&%#y1G%#y2G%"yG%"zG6"F-C$-T#6%9&8&8'-T%6&*&-8$6#F4 """-8%F;F<7$-%%subsG6$/F3F:9$-FA6$/F3F>9%7$F9F=F2F-F-6&%(getnameGc(getname%:-65 %,poltodiffeqGc)poltorec%:-356#/%+modulenameG%%gfunG M6R0 I%:-71R6$%%exprG%"xG6#%"iG6"F)@-4-%$hasG6$9$9%%%trueG-%%typeG6$F/%)functionG@%4 -F36$F/%'RootOfG-%&evalbG6#3-%)assignedG6#&T#6#-%#opG6$""!F/-F36$-FF6$-%%nopsG6 #F/F/<$-%'radfunG6$%)anythingGF0-%'algfunGFS-F36$-FF6$"""F/<$-FV6$FT%#_ZG-FRFhn -F36$F/FPF1-F36$F/<$%"*G%"+GC$?(8$FenFenFM-9!6$-FF6$FdoF/F0F)-F<6#/Fdo,&FMFenFe nFen-F36$F/%"^G553-F36$FY-%(polynomGFS4-F-6$-FF6$""#F/F03-F36$F[q%'posintG-Ffo6 $FYF03-F36$F[q%)rationalG-F36$FY-%)specfuncG6$FP%$expG%&falseGF)F)6$%,diffeqtab leGc,diffeqtable$:-9F) M6R0 I%:-23R6"6*%'resultG%(methodsG%'methodG%"uG%"nG%"sG%%unknG%%exprGF$F$C*@$0&9"6# """.%(stampedG-%'RETURNG6#-T%6#-T#6$""#F2>8+&F26#F@>8*&F26#""$>8%&F26#""%>8'-%# opG6$""!FF>8(-FR6#FF?&8&FK%%trueGC'>8)-%*traperrorG6#-T'6%F5FBFZ@$/Fhn%*lasterr orG\-%)userinfoG6'FI.%%gfunG%,Trying~the~GFZFhn>8$-T)6%FhnFVFP@$0Fjo%%FAILGC$-F do6(F@Ffo.%$TheGFZ.%1seems~to~satisfyGFjo-F86#7$FjoFZF`pF$F$6*%*typecheckGc*typ echeck&:-107%*listtorecGc*listtorec%:-23%+listtolistGc+listtolist%:-21%(l2r/l2r Gc(l2r/l2r%:-816#/%+modulenameGFgo M6R0 I$:-5R6"F$F$F$-T#6$%%trueG9"F$F$6$%.borelinvborelGc.borelinvborel%:-566#/%+modu lenameG%%gfunG M6R0 I%:-94R6%'%"sG%'seriesG%"xG%"oG6'%"pG%"vG%"kG%#pvG%$ppvG6"F0C(>8%*&9%"""-%#opG6 $F69$!"">8$-%(convertG6$F:%(polynomG>8&F6?(F0F6F6F01,&FC""#F6F69&C&>8'-T#6&F=F3 F5FF>8(-T%6&FKF3F5FF>F3,&F3F6-T'6&,&FKF6F5F;FPF5FFF;>FCFF@$2FCFHC%>FK-FM6&F=F3F 5FH>FP-FR6&FKF3F5FH>F3,&F3F6-FW6&FYFPF5FHF;-F'6%,&F3F6-%"OG6#)F5,&FHF6F6F6F6F5F ioF0F06(%+powcomposeGc+powcompose%:-90%/pprimeknowingpGc/pprimeknowingp%:-96%*p owdivideGc*powdivide%:-93F0 M6R0 I%:-46R6"F$F$F$@%0&9"6#""".%(stampedG-T%6#-T#6$""(F(-T)6&F+-T'6%F+&F(6#""#.%$og fG-%#opG6$""!F:&F(6#""$F$F$6*%*typecheckGc*typecheck&:-107%0seriestoratpolyGc0s eriestoratpoly%:-46%-seriestolistGc-seriestolist%:-45%.listtoratpolyGc.listtora tpoly%:-226#/%+modulenameG%%gfunG M6R0 I%:-79G6" M6R0 I%:-31"""6#/%+modulenameG%%gfunG M6R0 I%:-64R6"6$%"sG%"tGF$F$C%>8$-%-solve/linearG6#9">8%-%'selectG6%%%typeGF*%.gfun/ identityG@%0-%%nopsG6#F0"""%%FAILG-%%subsG6%/-%#opG6$F;-FBF:F;//F;F;F@F*F$F$F$F $ M6R0 I%:-12R6"6)%"yG%'resultG%"lG%"xG%(methodsG%#llG%"iGF$F$C.@$0&9"6#""".%(stampedG -%'RETURNG6#-T%6#-T#6$""#F1>8&&F16#F?>8$-%#opG6$""!&F16#""$>8'-FG6#FJ>8(&F16#"" %-%)userinfoG6%F3.%%gfunG%NTrying~to~find~a~rational~generating~functionG?&8*FR %%trueGC'>&8)6#Fgn-%*traperrorG6#-T'6$FAFgn@$/F[o%*lasterrorG\>8%-T)6&F4F[oFN7# .%$ogfG@$/Fio%%FAILGFgo-F76#7$,&*&-%&denomG6#&FioF2F3-FE6#FNF3F3-%&numerGFjp!"" Fgn-FW6%F3FY%PTrying~to~find~an~algebraic~generating~functionG?&FgnFRFhnC$>Fio- T+6&F4F[oF\qF]p@$0FioFbp-F76#7$F[qFgn-FW6%F3FY%NTrying~to~find~a~linear~differe ntial~equationG?&FgnFRFhnC$>Fio-T-Fiq@$F[rF\rFbpF$F$6.%*typecheckGc*typecheck&: -107%)guesseqnGc)guesseqn%:-12%+listtolistGc+listtolist%:-21%.listtoratpolyGc.l isttoratpoly%:-22%,listtoalgeqGc,listtoalgeq%:-18%-listtodiffeqGc-listtodiffeq% :-196#/%+modulenameGFZ M6R0 I&:-102R6$%*equationsG%)unknownsG6.%$eqnG%%eqnsG%%varsG%$solG%%solsG%"iG%"kG%"l G%&pivotG%$varG%"cG%"jG6"F4C)>8%<#-%$seqG6$*(8+"""-%)icontentG6#F=!""-%%signGFA F>/F=-%&minusG6$9$<#""!?(8*F>F>F40F7<"C+>F=-%$mapG6$%'lengthG7#-%#opG6#F7-%'mem berG6%-%$minG6#-FXFAF=.8)>8$&F76#F\o>8&-%*intersectG6$-%'indetsG6#F^o9%>8-&Fbo6 #F>>8,-%&coeffG6%F^oF[pF>?(8/""#F>-%%nopsG6#Fbo%%trueGC$>8.-Fap6%F^o&Fbo6#FdpF> @$2-FU6#F\q-FU6#F_pC$>F_pF\q>F[pF_q>F7-FG6$-%%subsG6$/F[p,$*&F_pFB-F^r6$/F[pFKF ^oF>FB-%'subsopG6$/F\o%%NULLGF7FJ>&8'6#FM6%F[pF^oF_p>8(FO>FdpFK?(F\o,&FMF>FBF>F BF>FipC%>&F^oF`o,$-F^r6%/&&F]sF`oF]pFKFas&F]t6#FepFB@$0FgsFKC$>Fdp,&FdpF>F>F>>F =-Fgr6$/FdpF\oF=>Fas-%&unionG6$Fas<#/F\t*&FgsF>&F]t6#""$FB>Fbo-FG6$<#-FX6#Fio<# -F:6$-FX6$F>F\o/F\oFas@)/FboFOFas/FfpF><$/-FXFhpF>-FX6#-F^r6$FcvFas0-FG6$-%'sel ectG6%%%typeG<#-F:6$-FX6$FepF\oF^v%)rationalGFJFO<$-F:6$/F\oFK/F\oFbo-FX6#-F^r6 $FgwFas%%FAILGF4F4F4F4 M6R0 I%:-87R6'%"aG%"bG%"yG%"sG%"tG6-%"qG%"rG%"uG%"vG%#duG%#dvG%&alphaG%%betaG%)oldal phaG%(oldbetaG%#ttG6"F6C&>%&OrderG%)infinityG>8&-%'seriesG6$9$9&>8'-F>6$9%FA@'/ F<""!FF/FCFIF@C*>8,""">8-FI>8*FI>8+FN>8(-%#opG6$-%%nopsG6#F8)-FX6$-Fen6#FCF C?(F6FNFNF6%%trueGC/@$2FVFhnC*>8.Fhn>FhnFV>FVFdo>FdoFC>FCF<>FFdo FR>FRFM>FMFdo@$2""%F]pC%>FdoFT>FTFP>FPFdo-%)userinfoG6'F\p.%%gfunG%;degrees~of~ the~polynomialsGFVFhn-Fjp6'""&F\q.%,polynomialsGF8$*(-FX6$,& FZFN!""FNF8%-%'normalG6#-F>6$,&FFdo,&FMFN*&FjqFNFRFNF_r>FMFR>FRFdo@$FcpC%>Fdo ,&FPFN*&FjqFNFTFNF_r>FPFT>FTFdo>FFVFhn>FCFfr>FhnFin@%FgqC%@$F[p>9'-%(conver tG6$-Fhr6#-F>6$*&FRFN-%&coeffG6%FCFAFIF_rFA%(polynomG@$Fcp>9(-Fht6$-Fhr6#-F>6$* &FTFNF_uF_rFAFbuFNC%@$F[p>Fft-Fht6$-Fhr6#-F>6$FRFAFbu@$Fcp>Feu-Fht6$-Fhr6#-F>6$ FTFAFbuFCF6F6F6F6 M6R0 I%:-54G6" M6R0 I%:-39R6&%%exprG%%aofnG%%foftG'%$iniG%(booleanG6*%"rG%"aG%"nG%"fG%"tG%)iniconds G%"AG%"NG6"F4C(-T#6%9&8'8(@%5/9#""$9'>8$-T%6&7$9$9%.8%.8&.8)>FC-FE6%FGFJFL@$F=> FO-T'6'FCFKFMFO%&falseG>FC-%%subsG6$7$/FK8*/FM8+FC@%F=>FO-Fen6$FgnFO>FO%%NULLG- T+6(-T)6$FCF[oFinF[oF:F;FOF4F46,%(getnameGc(getname%:-65%*formatrecGc*formatrec %:-62%3goodinitvalues/recGc3goodinitvalues/rec%:-67%-listprimpartGc-listprimpar t%:-73%1rectodiffeq/doitGc1rectodiffeq/doit%:-986#/%+modulenameG%%gfunG M6R0 I%:-72R6%%"AG%"uG%"xG6)%"kG%"iG%"lG%"vG%$relG%$unkG%"jG6"F0C)>8$-%%nopsG6#9%@$0 -&%'linalgG6#%'rowdimG6#9$F3-%&ERRORG6$%9incorrect~number~of~rowsG-%#opGF?>8&-& F<6#%'coldimGF?-%)userinfoG6)""#.%%gfunG%Elooking~for~a~linear~dependency~in~aG F3.9&FH.%'matrixG>8)7#-%$seqG6$&8'6#8%/F[o;"""F3>8(-T#6%-&F<6#%(augmentG6$F@-&F <6#FW6%F3F^oFYF3,&FHF^oF^oF^o@%/F`o%%FAILGF`pC$>F`o-%)primpartG6$-%&numerG6#-%( convertG6$7#-Ffn6$*&-%'normalG6#-%%subsG6$7$/F[oF^o-Ffn6$/8*""!/F[r-Fdq6$/F[o%% NULLGFYF`oF^oF[oF^o/F[oFY%"+GFY-Fjp6$7#-Ffn6$*&FcqF^o&F76#-FFFjnF^oFbrFcrF0F06$ %/cheapgausselimGc/cheapgausselim%:-57F0 M6R0 I%:-24R6"F$F$F$@%0&9"6#""".%(stampedG-T%6#-T#6$F*F(-%$mapG6$%'normalG-&T'6#&F(6 #""%6$&F(6#""#&F(6#""$F$F$6(%*typecheckGc*typecheck&:-107%-listtoseriesGc-listt oseries%:-24%2listtoseriestableGc2listtoseriestable%:-256#/%+modulenameG%%gfunG M6R0 I%:-57R6%%"AG%%nlinG%%ncolG6)%&lcolsG%"cG%"kG%"iG%"uG%"jG%$pivG6"F0C%>8$7#-%"$G 6#;"""9&?(8%F9F99%%%trueGC*?&8&F3/-%'normalG6#&9$6$FFFFI@$/FAF:-%'RETURN G6#&FG6$F8*FF?(8',&FC$@%/&FG6$FUFAFI\>8(FZ?&8)F3F>>&FG6$FUFjn- FD6#,&*&FSF9F\oF9F9*&FhnF9&FG6$F>&FG6$FF3-%%subsG6$/FA%%NULLGF3%%FAILGF0F0F0F0 M6R0 I$:-4R6''%$PolG%(polynomG'%"xG%%nameG'%"yGF*'%$ordG%*nonnegintG%9optional_posit ive_slopesG6"F1F1-%$mapG6%T#-T%6#9"9%F1F16&%:algeqtoseries/prettyprintGc:algeqt oseries/prettyprint%:-50%3algeqtoseries/doitGc3algeqtoseries/doit%:-496#/%+modu lenameG%%gfunG M6R0 I%:-95R6%%$polG%"xG%"nG6#%"iG6"F*@'1-%'degreeG6$9$9%9&F02F2-%(ldegreeGF/""!-%(c onvertG6$7#-%$seqG6$*&-%&coeffG6%F0F18$""")F1FBFC/FB;F6F2%"+GF*F*F*F* M6R0 I%:-47R6"F$F$F$@%0&9"6#""".%(stampedG-T%6#-T#6$""'F(-T)6&F+-T'6%F+&F(6#""#.%$og fG&F(6#""$&F(6#""%F$F$6*%*typecheckGc*typecheck&:-107%,seriestorecGc,seriestore c%:-47%-seriestolistGc-seriestolist%:-45%*listtorecGc*listtorec%:-236#/%+module nameG%%gfunG M6R0 I%:-80G6" M6R0 I%:-327$%$ogfG%$egfG6#/%+modulenameG%%gfunG M6R0 I%:-65R6%'%%yofzG-%)functionG6#%%nameG%"yG%"zG6"F-F-C%>9%-%#opG6$""!9$@$-%%type G6$F0%*procedureG-%&ERRORG6$%7not~an~unassigned~nameGF0>9&-F26#F5F-F-F-F- M6R0 I%:-17c)invborel%:-166#/%+modulenameG%%gfunG M6R0 I%:-11R6&%$eqnG%%yofzG%%uofkG'%$iniG%(booleanG6*%)inicondsG%"fG%"yG%"zG%"uG%"kG %"YG%"ZG6"F4C'-T#6%9&8(8)@%5/9#""$9'>8%-T%6&7$9$9%.8&.8'.8$>FC-FE6%FGFJFL>FC-%% subsG6$7$/FK8*/FM8+FC@%F=>FO-FU6$FXFO>FO%%NULLG-T'6(FCFYFenF:F;FOF4F46(%(getnam eGc(getname%:-65%-formatdiffeqGc-formatdiffeq%:-60%1diffeqtorec/doitGc1diffeqto rec/doit%:-586#/%+modulenameG%%gfunG M6R0 I&:-103R6$%$RecG%%uofkG6,%$recG%"uG%"kG%$iniG%"nG%"iG%"cG%#dcG%#coG%"jG6"F2C$>8 $-T#6&7#9"8%8&8'@'/&F56#"""""!9$-%$hasG6$-%'subsopG6$/FB%%NULLGF5F<-%&ERRORG6$% 3invalid~recurrenceGFDC*>8(-%%nopsG6#F5>8*-%(collectG6$F@F<>8+-%'degreeG6$FXF<@ $0F=<">F=-T%6(F5F;F&8,6#!""FC?(8)FCFB,&FSFBFdoFBFbo> &Fgo6#F[p&F56#,&F[pFB""#FB?(F[pFCFBFgnFboC$>&Fgo6#,(FSFBF[pFBFioFB&Fgo6#,(FSFBF [pFBFdoFB?(8-F\qFioFCFbo>&Fgo6#F^q,&&Fgo6#,&F^qFBFioFBFBF`qFio-T'6&7$FC-%$seqG6 $F`q/F^q;FC,(FSFBFgnFBFioFBF;F8'-%(collectGF2>8(-%)p rimpartG6$FhnF4@$0-%'expandG6#,&F\oFGFhn!""""!-%'RETURNG6#-FY6$F\oF4>8&-%'degre eGF_o@)/F^pFGC$>Fhn,$*&-%&coeffG6%FhnF4FgoFG-Fhp6%FhnF4FGFfoFfo@'-F76$FhnFL<#Fh n45-F16$Fhn%'RootOfG-F16$Fhn%(&RootOfGF5C$>Fhn-%&evalaG6#-%'NormalG6#Fhn@%F]qF_ qF5/F^p""#C%>Fhn7#-FB6$-Fhp6%FhnF4FE/FE;FgoFbr>F^p-%%sqrtG6#,&*$-FT6$FbrFhnFbrF G*&-FT6$""$FhnFG-FT6$FGFhnFG!"%-F>6%F7-FW6$%'normalG<$,$*&,&F^pFGFbsFGFGFesFfo# FfoFbr,$*&,&F^pFGFbsFfoFGFesFfo#FGFbrFLFMFPC$>8$-%*traperrorG6#-%'isolveG6$-Fco FKF4@%/F[u%*lasterrorGF5<#-FB6$-FT6$FbrFE/FE-F>6%F7-FW6$FT7#F[u/-%*identicalG6# F4FLF-F-6$%)myisolveGc)myisolve%:-88F- M6R0 I%:-55G6" M6R0 I%:-15R6$%%exprG%%yofxG62%"yG%"xG%%deqsG%(newexprG%%funsG%#nfG%"iG%"jG%*sortbyl enG%#eqG%&initsG%%powsG%#npG%$iniG%$deqG%%expsG6"F8C5-T#6%9%8$8%@&-%$hasG6$9$F> -%&ERRORG6%%Bfirst~argument~should~not~containGF>FD4-T%6$FDF?-FF6$%\pexpression ~is~not~holonomic~(or~involves~a~function~which~is~not~implemented~yet)GFD@$4-F BFL-%'RETURNG6#,&F="""FD!"">8'FD>83-%'indetsG6$FZ)-%)specfuncG6$%)anythingG%$ex pG%)rationalG@$0Ffn<">FZ-%%subsG6$7#-%$seqG6$/8*-F_o6#*&-%#opG6$""#F]pFW-Fbp6#- Fbp6$FWF]pFW/F]pFfnFZ>8(7#-Fbp6#-%'removeG6%%%typeG-Fhn6$FZ%)functionG%'RootOfG >8)-%%nopsG6#F[q>8&-%$mapG6%T'F[q-F>6#F?>F]r-%&tableG6#7#-Fjo6$-Ffo6$/F>&F>6#F] p&F]rF_s/F]p;FWFhq>FZ-Ffo6$7#-Fjo6$/&F[qF_s-F^sFcrFasFZ>8/-F`q6%Fbq-Fhn6$FZ)-%( polynomG6$F^oF?-%*gfun/freeGFcr)F^oF`o>80-Fjq6#F]t?(F]pFWFWFjt%%trueGC%>81-Ffo6 $/F?""!-Fbp6$FW&F]tF_s>82,&*&FfuFW-%%diffG6$-&F>6#,&F]pFWFhqFWFcrF?FWFW*(-Fbp6$ FdpFhuFW-F^v6$FfuF?FWF`vFWFX@%/FauFeu>&F]rFbvFju>F\w<$/-Fav6#Feu)FauFevFju>FZ-F fo6$7#-Fjo6$/FhuF`v/F]p;FWFjtFZ>F[q-F`q6%Fbq-Fhn6$FZ<$-%'radfunGFet-%'algfunGFe tFct>8,R6$%"aG%"bGF86$%)operatorG%&arrowGF8-%&evalbG6#1-%'lengthG6#F=-Fdy6#FDF8 F8F8?(F]pFWFWF80F[qFcoC'>F[q-%%sortG6$7#-FbpF[rFgx>8--T)6&&F[q6#FWFbr.8..%*alge braicG>&F]r6#,(F]pFWFhqFWFjtFW-Ffo6$/F>&F>F^[l-T+6%FbzFbrFiz>FZ-Ffo6$/Ffz-Fc[lF crFZ>F[qF]x-T-6&FZ7#-Fjo6$&F]r6#8+/Fe\l;FW,*FhqFWFjtFWF]pFWFXFW7#-Fjo6$-&F>Fd\l FcrFf\lFbrF8F86.%(getnameGc(getname%:-65%,isholonomicGc,isholonomic%:-71%,funto diffeqGc,funtodiffeq%:-63%.algfuntoalgeqGc.algfuntoalgeq$:-1%.algeqtodiffeqGc.a lgeqtodiffeq$:-3%,poltodiffeqGc,poltodiffeq%:-346#/%+modulenameG%%gfunG M6R0 I%:-40R6$%$RecG%%uofkG6*%$recG%"uG%"kG%$iniG%"nG%"iG%"cG%#dcG6"F0C$>8$-T#6&7#9" 8%8&8'@%/&F36#"""""!9$C'>8(-%%nopsG6#F3@$0F;<">F;-T%6(F3F9F:F;%%trueG,&FEF@!"#F @>8*F>>8+,$-%%subsG6$/F:,&F:F@F@F@FT!""-T'6&-%$mapG6%%(collectG7&FA*&FVF@&F36#" "#F@-%$seqG6$,&*&FVF@&F36#8)F@F@*&FTF@-FY6$Fen&F36#,&F[pF@FgnF@F@F@/F[p;""$FE*& FTF@-FY6$Fen&F36#FEF@F:F9F:F;F0F06(%*formatrecGc*formatrec%:-62%3goodinitvalues /recGc3goodinitvalues/rec%:-67%(makerecGc(makerec%:-846#/%+modulenameG%%gfunG M6R0 I%:-73R6$%"lG%"zG6%%"iG%"TG%"qG6"F+@%-%(hastypeG6$9$%&floatGF0C$-%(contentG6%-% (convertG6$7#-%$seqG6$*&-%#opG6$8$F0""")8%,&FAFB!""FBFB/FA;FB-%%nopsG6#F0%"+GFD .8&7#-F;6$-%&coeffG6%FNFDFA/FA;""!,&FIFBFFFBF+F+F+F+ M6R0 I%:-25=6"%&falseGE\[l(%$egfGR6$%"lG%"xG6#%"iGF$F$-%'seriesG6%-%(convertG6$7$-%$ seqG6$*(-%#opG6$8$9$""")9%,&F!""F>F>-%*factorialG6#FAFB/F<;F>-%%nopsG6#F=-%" OG6#)F@FH%"+GF@FHF$F$F$%'lgdegfGR6$%"LGF+6%F*F-%#nlGF$F$C'>FF>F$30F<7" /&F<6#F>""!>F<-%'subsopG6$/F>%%NULLGF<@$/F8&-FI6#F<-F/6%*&-F26$7#-F66$*(&F<6#8%F>)F@,&FepF>!"#F>F>-FD6#FgpFB/Fep;""#F goFOF>-F26$7#-F66$*(FcpF>)F@,&FepF>FBF>F>-FD6#FeqFB/Fep;F>FgoFOFBF@,&FgoF>FBF>F $F$F$%'revegfGRFRFTF$F$C'>FF>F$FY>FFgoFho-T#6%-F/6%-F26$7 $-F66$*(FcpF>)F@FepF>-FDFdpFBFhq)F@,&FgoF>F>F>FOF@FasF@FgoF$F$6$%*powrevertGc*p owrevert%:-94%'revogfGRFRFTF$F$C'>FF>F$FY>FFgoFho-Fdr6%-F /6%-F26$7$-F66$*&FcpF>F^sF>FhqF`sFOF@FasF@FgoF$F$Fbsc(Laplace%:-17RF)F,F$F$-F/6 %-F26$7$-F66$*(F9F>F?F>FCF>FFFKFOF@FHF$F$F$%'lgdogfGRFRFTF$F$C'>FF>F$F Y>FFgoFho-F/6%*&-F26$7#-F66$*(FepF>&F<6#,&FepF>F>F>F>FdqF>/Fep;F>F jqFOF>-F26$7#-F66$*&FcpF>FdqF>FhqFOFBF@FjqF$F$F$%$ogfGRF)F,F$F$-F/6%-F26$7$-F66 $*&F9F>F?F>FFFKFOF@FHF$F$F$6#/%+modulenameG%%gfunG M6R0 I%:-58R6(%"RG%"yG%"zG%"uG%"kG%)inicondsG60%"lG%$iniG%"iG%$recG%"jG%*minordrecG% *maxordrecG%"mG%"rG%$dr1G%'inhdegG%(inhpartG%"pG%#rrG6"F:C0@&-%$hasG6$9$9(-%&ER RORG6$FA%Kcannot~appear~in~the~differential~equationG-F>6$F@9'-FC6$FHFE@%/9#""' C&>8$<#-%$seqG6$-%#opG6$""#-FW6$""!-FW6$Ffn8&/Fin-%&minusG6$-%'indetsG6$7$9)F@- %,gfun/initeqG6#9%<$-Ffo6#Ffn--%"DGFeoFio>8%7%/Fho-FHFio/Fjo-FH6#"""-FT6$/---%# @@G6$F\pFinFeoFio*&-FH6#FinFep-%*factorialGF`qFep/FinFQ>8,-%%subsG6$F^pF@>F^p-F gq6$F^pFboC$>FeqF@>F^p<"@$0&FeqFdpFfn?(8.FfnFepF:%%trueGC&>80,&FepFep9&!""?(Fin FYFep-%%nopsG6#Feq/-%'degreeG6$FhrFjrFep>Fhr-%$gcdG6$Fhr&FeqF`q@$4-F>Fcs[>Feq7$ Fbr-FT6$-%$quoG6%FhsFirFjr/Fin;FYF]s>81-%$mapG6$R6#%"xGF:F:F:@%/F@FfnFepF@F:F:F :Feq>8)-%$minG6#-FT6$,&FinFep-Fbs6$-FW6$,&FinFepFYFepFgtFjrF[s/Fin;Ffn,&F]sFep! "#Fep>8*-%$maxG6#-FT6$,&FinFep-%(ldegreeGFiuF[sF]v>8'-%&arrayG6$%'sparseG;FauFb v?(FinFYFepF]sFer?(8(-Fjv6$-FW6$FinFeqFjrFep-FbsFfwFer>&F\w6#,(FinFepF`vFepFdwF [s,&F[xFep*&-%&coeffG6%FgwFjrFdwFep-%'expandG6#-%(convertG6$7#-FT6$,&FAFep8+Fep /F]y;,&FepFepFdwF[sF]x%"*GFepFep@%/FbrFfnC$>8-F[s>8/FfnC$>Ffy-Fbs6$FbrFjr@%0Fdr FfnC$>Fhy*&-Fdx6#-Fgx6$7#-FT6$*&-Fax6%FbrFjrFinFep-Fgx6$7#-FT6$,&FAFepFdwFep/Fd w;,&FepFepFinF[s,(FdrFepFinF[sF[sFepFayFep/Fin;FfnFfy%"+GFep-Fbq6#,&FdrFepF[sFe pF[s>Feq-%'subsopG6$/Fep-%'seriesG6%*&FbrFep)FirFdrF[sFjr,&-Fdv6$Ffy,&FbvFepFau F[sFepFepFepFeq>FhyFfn@$50F^pF_rFLC$>F^p<$-FW6#F^p-FW6#-Fit6%Fgx7#-FT6$-Fgq6$/F AFin7$Fjz-FT6$*&&F\w6#FdwFep-FH6#,&FinFepFdwFepFep/Fdw;-Fdv6$Fau,$FinF[sFbvFf[l Fh[l?(Fin,&FfyFepFepFepFepF:52Fin,$FauF[s/-Fgq6$F\^l<#-FT6$&F\w6#,&FbvFepFdwF[s /Fdw;Ffn,(FinFepFfyF[sF[sFep<#Ffn>F^p<$Fa]l-Fgx6$-Fgq6$F\^l7$F^^lFhyFh[l>F\w-T# 6$-Fgq6$/FA,&FAFepFauF[s7$Fhy-FT6$&F\wF`q/FinFawFA?(F:FepFepF:32FY-F^s6#F\w/&F\ w6#FdalFfn>F\w-F^\l6$/Fdal%%NULLGF\w@$F\]l>F^p-T%6'F\wFHFAF^pFer-T'6&-Fit6%%(co llectGF\wFAFHFAF^pF:F:6(%-listprimpartGc-listprimpart%:-73%3goodinitvalues/recG c3goodinitvalues/rec%:-67%(makerecGc(makerec%:-84F: M6R0 I%:-14R6%%$eq1G%$eq2G%%yofzG6+%"uG%#u1G%#u2G%"nG%#r1G%#r2G%%indsG%"iG%"jG6"F2C' >8(-T#6%9$9&-8%6#8'>8)-F76%9%F:-8&F=>8*-%*intersectG6$-%'indetsG6$F5&%#_CG6#%)a nythingG-FL6$F@FN@$0FG<"C$>8,-%$maxG6#-%#opG6#-%$mapG6$Fhn-%&unionGFJ>F@-%%subs G6$7#-%$seqG6$/&FG6#8+&FO6#,&Fjo"""FYF^p/Fjo;F^p-%%nopsG6#FGF@-T'6%-T%6&*&F;F^p FDF^p7$F5F@7$F;FD-8$F=F]qF:F2F26(%,diffeqtorecGc,diffeqtorec%:-11%)poltorecGc)p oltorec%:-35%,rectodiffeqGc,rectodiffeq%:-396#/%+modulenameG%%gfunG M6R0 I$:-3R6"60%"yG%"zG%"PG%&initsG%"gG%"uG%"dG%"iG%"YG%$deqG%"jG%"rG%&inityG%#P0GF$ F$C1>8&-T#6&7#9".8$.8%.8'>8(-T%6&-%%diffG6$F6F=F6F=.8)@$-%$hasG6$FCF=-%'RETURNG 6#-T'6%-%'normalG6#*&F6"""FC!""-F=6#F?FA>8*-%'degreeGFI-%)userinfoG6&""$.%%gfun G%+degree~is~GFin@'1FinFZ>8--%%subsG6$/F=FfnF64-FN6$F6F?>Ffo,&FfnFZ-%'RootOfGFI FenC%>&8,6#FZ-%$remG6%,$*(FKFZFCFen-FHF]pFZFenF6F=?(8+""#FZ,&FinFZFenFZ%%trueG> &Fep6#F^q-Fhp6%-%(collectG6$,&-FH6$&Fep6#,&F^qFZFenFZF?FZ*&-FH6$F]rF=FZFdpFZFZF =F6F=>Ffo-T)6%-%&arrayG6%;FZ,&FinFZFZFZ;FZFin-FW6#7%7$FZ-%"$G6$""!F`q7%FdsFZ-Fb s6$Fds,&FinFZ!"#FZ-%$seqG6$7#-F[t6$-%&coeffG6%FcqF=8./Fct;FdsF`q/F^q;FZF`q7$FZ- F[t6$-FH6$Ffn7#-Fbs6$F?F^q/F^qFetF?-F]o6&FZF`o%9differential~equation~isGFfo>81 -FW6#-Fho6&/F?Fds/F=-F=6#FdsFAF6@$34-FN6$FeuF=0FeuFds@%/FA<"-FQ6#Ffo-%&ERRORG6# %;invalid~initial~conditionsG@$Fcv@$0-%*traperrorG6#-%&evalaG6#-Fap6#-Fho6$/F\v %#_ZGFeu%*lasterrorG>FA-%&unionG6$FA<#/F\vFfw>80/F=-Fho6$FAF\v>FA-F_x6$<#/F\v-% #opG6$F_qFdx-Fho6$F\yFA?(F^qFZFZF`qFaqC$>8/-Faw6#/---%#@@G6$%"DGF^q6#F=F]v-Fho6 $7$FjuFdxFcq@%0FeyF\x>FA-F_x6$FA<#Fey[>FA-T-6&-T+6#7$FfoFfnF=F?-%'selectG6%%%ty peGFA/%)anythingG-%*gfun/freeG6#F\v@%FevFgv-FQ6#<$-F^y6#FAFfoF$F$6.%,formatpole qGc,formatpoleq%:-61%(mygcdexGc(mygcdex%:-87%.algeqtodiffeqGc.algeqtodiffeq$:-3 %'lindepGc'lindep%:-72%-formatdiffeqGc-formatdiffeq%:-60%6goodinitvalues/diffeq Gc6goodinitvalues/diffeq%:-666#/%+modulenameGFao M6R0 I%:-96R6&%%poffG%"fG%"xG%"nG6"F)F)-T#6&-%%diffG6$9$9&-F.6$9%F1F19'F)F)6$%*powdi videGc*powdivide%:-93F) M6R0 I%:-16R6"F$F$F$-T#6$%&falseG9"F$F$6$%.borelinvborelGc.borelinvborel%:-566#/%+mo dulenameG%%gfunG M6R0 I%:-48R6"F$F$F$@%0&9"6#""".%(stampedG-T%6#-T#6$"")F(-T)6&F+-T'6%F+&F(6#""#.%$og fG-%#opG6$""!F:&F(6#""$F$F$6*%*typecheckGc*typecheck&:-107%/seriestoseriesGc/se riestoseries%:-48%-seriestolistGc-seriestolist%:-45%-listtoseriesGc-listtoserie s%:-246#/%+modulenameG%%gfunG M6R0 I%:-81R6%%"lG%"nG%"uG6.%"iG%"sG%"zG%"jG%#nbG%%lpolG%)ordereqnG%'resultG%"kG%%lv arG%*nbinicondG%)inicoeffG6"F5C&>8(-%%nopsG6#9$>8%7$-T#6$F<8&-%$seqG6$-FA6$7#-F E6$*&&F<6#8'"""),&FOFP!""FP8$FP/FO;FPF8FC/FT;FP,&T%FPFSFP?(FT-%$maxG6$FPFZFP-%$ minG6$T',&F9FPFSFP%%trueGC&>F>7$-%#opG6#F>FG-%)userinfoG6&""%.%%gfunG%SLooking~ for~recurrence~with~coefficients~of~degreeGFT>8)-T+6(F>FC-Fjn6$F8*&,&FTFPFPFPFP ,&T)FPFPFPFPF8%&falseG,&F8FP""#FP@$0F]p%%FAILGC)>8*-Fgn6#-Fco6#-%$mapG6%%'degre eGF]pFC>8-7#-FE6$-9&6#,&9%FPFOFP/FO;""!F_q>8+-%(convertG6$7#-FE6$*&-Fhr6$7#-FE6 $*&-%&coeffG6%&F]p6#8,FCFOFP),(FarFPF_qFPFOFS,&FisFPFSFPFP/Fis;FPFdp%"+GFP-F^r6 #F[tFPFbrF_t>8/-%%subsG6$/Far,&FarFPF_qFS-Fes6%Ffr-F^r6#,&FarFPF_qFPFP>8.,&-Fgn 6$F_q-T-6$FctFarFPFSFP>Ffr-%(collectG6$-%)primpartG6$FfrFiqFiq-%'RETURNG6#<$-FE 6$/-F^rFN&F<6#,&FOFPFPFP/FO;Fdr-Fjn6$F_u,&F8FPFSFPFfrF\qF5F56.%-listtoseriesGc- listtoseries%:-24%,mindegcoeffGc,mindegcoeff%:-29%,maxdegcoeffGc,maxdegcoeff%:- 26%,maxordereqnGc,maxordereqn%:-28%.guessandcheckGc.guessandcheck%:-68%-firstno nzeroGc-firstnonzero%:-59F5 M6R0 I%:-33R6&'%-functionlistG-%%listG6#%*algebraicG'%&pointG<$%%nameG/F.F*'%)accura cyG<$%(integerG-F(6#%*nonnegintG'%)optionalG-%*identicalG6#%%easyG6+%"xG%"aG%"n G%"lG%"iG%$accG%"mGF<%'resultG6"FFC(@%-%%typeG6$9%%"=GC$>8%-%#opG6$""#FL>8$-FR6 $"""FLC$>FP""!>FVFL>8&-%%nopsG6#9$@%-FJ6$9&F(C%>8*-%$maxG6#-FR6#Fao>8'7#-%$seqG 6$,&FdoFY8(!""/FapFao>8),(-%(convertG6$Fao%"+GFYFinFYFbpFYC$>F[p7#-%"$G6$FfnFin >FepFao>8+-%&evalbG6#/9#""%>8,-T#6'-%$mapG6&%'taylorG-%%subsG6$/FV,&FVFYFPFYF]o FVFepFVF[p,&FepFYFbpFYFcq@%/F[r%%FAILGF[s-Fdr6$/FV,&FVFYFPFbpF[rFFFF6$%+pade2/d oitGc+pade2/doit&:-1006#/%+modulenameG%%gfunG M6R0 I%:-66R6'%$deqG%"yG%"zG%$iniG%"pG61%"uG%%initG%"iG%$solG%)maxorderG%$ordG%%inds G%"jG%$rejG%"kG%&inds2G%$sysG%"vG%#gbG%&dorejG6"F:C*>8(-T#6%9$9%9&@$/9#""&>F=-% $maxG6$F=9(>8)F=@%32""$FF-%%typeG6$9'%$setGC%>8%FV>F=-FJ6$F=-%$seqG6$-%#opG6$"" #-F\o6$""!-F\o6$Fao8&/Fdo-%&minusG6$-%'indetsG6$FZ-%,gfun/initeqG6#FB<$-FB6#Fao --%"DGF^pFap@$31F=Fao-%$hasG6$FZFbp>F=""">FZ<">8$-%'seriesG6%-%%evalG6#-%%subsG 6$/-FB6#FC-Fhq6$FZ-%(convertG6$7$-Fin6$*(---%#@@G6$FdpFdoF^pFapF\q)FCFdoF\q-%*f actorialG6#Fdo!""/Fdo;FaoF=*&-%"OG6#F\qF\q)FC,&F=F\qF\qF\qF\q%"+G-T%F@FC%)infin ityG>F`q-%&unionG6$-Fgo6$<#-Fin6$-%&coeffG6%F`qFCFdoF`s<#FcsFZ>8'-%&solveG6$F`q -%*intersectG6$<#-Fin6$FfrF`s-Fjo6$F`q-%)functionGFap@'/Fjt%%NULLG@'30FNF_s0-Fh q6$/FCFao-F\o6$-%%nopsG6#FAFAFao-%&ERRORG6#%82-%&evalbG6#0-Fhq6$Fav&FA6#FdvFao@%F^w>Fjt-Fhq6$Fjt7#-Fin6$Ffr/F do;FaoFN>Fjt-Fhq6$Fjt7#Fbu>8*-%'selectG6%Fip-Fjo6$FjtFguFau>8+-FJ6#-F\o6#-%$map G6$F\o-Fjo6$7$FAFjt&%#_CG6#%)anythingG@$/F\y,$F[tF_s>F\yF_s?&FdoFex%%trueG@%3-% 'memberG6%FdoFjt.8-2F\q-Fev6#-Fgx6%FipFjtFdoC%>F\y,&F\yF\qF\qF\q>Fjt-Fhq6$/Fdo& Fhy6#F\yFjt>&8,F^sFju>Fg[lFgz>8.-Fgo6$<#-%"$G6#;F\q-Fev6#Fjt<#-Fin6$Fg[l/FdoFex @%33334-%(hastypeG6$FAFgy-F`]l6$-%'removeG6%FT7#-Fin6$&FjtF^s/FdoF[\l%%nameGFgy 2-FJ6#-F\o6#-Fby6%%'degreeGFjt-Fjo6$FjtFgyFR4-Fip6$Fjt%'RootOfG-FT6$Fg]l-%%list G6#-%(polynomG6$%)rationalGFe^lC(>8/<#-Fin6$,&&80F^sF\qFj]lF_sF[^l>Fex-Fjo6$Ff_ lFgy>F[\l-%%sortG6#7#-F\o6#F[\l>81<#-F\o6#-Fe]l6%F`]l-&%(grobnerG6#%'gbasisG6%F f_l7$-F\o6#Fex-Fin6$F[`lF[^l.%%plexGFgy@$-FT6$Fh`l-FW6#-%'linearG6#7#FgalC%>Fjt -Fhq6$-F\u6$Fh`l<#FgalFcbl>F\yF_s?(FdoF\qF\qFc\lF`z@$/Fj]l&F\`l6#&F[\lF^s@%3/-F ev6#-Fgx6%FipFjtFj]lF\q1FdoF=>Fg[lFdoC%>Fg[lFju>F\yF_[l>Fjt-Fhq6$/F_clFd[lFjt<# -Fin6$/---Fir6$Fdp,&FaclF\qF_sF\qF^pFapFj]l/Fdo-Fgo6$F^\l<#-Fin6$Fg[l/FdoFb\l<# -Fin6$/---Fir6$Fdp,&FdoF\qF_sF\qF^pFapFj]lF[^lF:F:6&%*nbinicondGc*nbinicond%:-8 9%+makediffeqGc+makediffeq%:-83F: M6R0 I%:-18R6"6+%'resultG%#exG%(methodsG%'methodG%"yG%"xG%"sG%%unknG%%exprGF$F$C+@$0 &9"6#""".%(stampedG-%'RETURNG6#-T%6#-T#6$""#F3>8,&F36#FA>8+&F36#""$>8&&F36#""%> 8(-%#opG6$""!FG>8)-FS6#FG>8%FC?&8'FL%%trueGC'>8*-%*traperrorG6#-T'6&F6FenFWFgn@ $/F[o%*lasterrorG\-%)userinfoG6'FJ.%%gfunG%,Trying~the~GFgnF[o>8$-T)6%F[oFWFQ@$ 0F]p%%FAILGC$-Fgo6(FAFio.%$TheGFgn.%1seems~to~satisfyGF]p-F96#7$F]pFgnFcpF$F$6* %*typecheckGc*typecheck&:-107%,listtoalgeqGc,listtoalgeq%:-18%-listtoseriesGc-l isttoseries%:-24%(s2a/s2aGc(s2a/s2a&:-1056#/%+modulenameGFjo M6R0 I%:-10R6$%$DeqG%%yofxG6*%$deqG%"yG%"xG%$iniG%"cG%#dcG%"nG%"iG6"F0C$>8$-T#6&7#9" 8%8&8'@%/&F36#"""""!9$C'>8*-%%nopsG6#F3@$0F;<">F;-T%6'F3F9F:F;,&FEF@!"#F@>8(F>> 8),$-%%diffG6$FSF:!""-T'6&-%$mapG6%%(collectG7&FA,&*&FUF@&F36#""#F@F@*&FSF@-FX6 $F_oF:F@F@-%$seqG6$,(*&FUF@&F36#8+F@F@*&FSF@-FX6$FjoF:F@F@*&FSF@&F36#,&F\pF@FZF @F@F@/F\p;""$FE*&FSF@&F36#FEF@F:F9F:F;F0F06(%-formatdiffeqGc-formatdiffeq%:-60% 6goodinitvalues/diffeqGc6goodinitvalues/diffeq%:-66%+makediffeqGc+makediffeq%:- 836#/%+modulenameG%%gfunG M6R0 I&:-104R6%%"sG%"xG%"yG6+%&serieG%"iG%%bigOG%%lpolG%"jG%'maxordG%$nbzG%"kG%+tryf actorsG6"F2C(@%0-%#opG6$,&-%%nopsG6#9$"""!""F>F=-%"OG6#F>>8&-F76$F:F=>FD,&FEF>F ?F>>8*-F76$""#F=>8,-%%typeG6$-%%subsG6$/F@""!7#-F7F<-%%listG6#%)rationalG>8)-%$ minG6$,&FDF>F?F>T#?(8%T%F>Fin%%trueGC'>8$7%F>F=-%$seqG6$-%%diffG6$F=-%"$G6$9%8+ /Fap;F>F`o-%)userinfoG6&""%.%%gfunG%KLooking~for~differential~equation~of~order GF`o>8'-T)6(FeoF`p-F[o6$,(FDF>F`oF?F>F>*&,&T'F>F>F>F>,&F`oF>FMF>F>,(FDF>F`oF?FJ F?FO,&FDF>FMF>@$0F\q%%FAILG-%'RETURNG6#-%(convertG6$7%&F\qFB*&&F\q6#FMF>-9&6#F` pF>-Fho6$*&&F\q6#8(F>-F[p6$Fgr-F^p6$F`p,&F_sF>!"#F>F>/F_s;""$Ffq%"+G@$/F`oFin-F ]r6#F[rF[rF2F26*%,maxordereqnGc,maxordereqn%:-28%,minordereqnGc,minordereqn%:-3 1%,maxdegcoeffGc,maxdegcoeff%:-26%.guessandcheckGc.guessandcheck%:-68F2 M6R0 I%:-89R6%%$deqG%"yG%"zG6$%$polG%$valG6"F+C$>8$-T#6&9$9&F38%@)/&F26#"""""!,&-T%6 $F.F3F9!""F9/-%%subsG6$/F3,&-%(ldegreeG6$F7F3F9F4F?F.F:@%/-FB6$/F3F:&F26#-%%nop sG6#F2F:F:-%&ERRORG6#%8'<$/*&F8"""-8%6#F8F@-FB6#,&F8F@!""F@/-FB6#""!F@>F<<$FH/FA*&F8F@FDF@-T%6&*&- F7FCF@FAF@7$9%F<7$FTFAFT0&9"6#""%.%'diffeqG-%&ERRORG6$%1invalid~argumentGFY-T)6 %-9!6%F:-T'6%FVF6FTFTFTF6F-F-6*%(getnameGc(getname%:-65%)poltorecGc)poltorec%:- 35%,diffeqtorecGc,diffeqtorec%:-11%,rectodiffeqGc,rectodiffeq%:-39F- M6R0 I%:-41R6$%%exprG%%yofnG6B%"TG%"YG%"NG%"RG%&INITSG%(ARGLISTG%(RECSTATG%"AG%"UG%" IG%&REMBRG%&ORDERG%#N0G%"nG%"aG%"lG%$resG%"iG%(INIVECTG%'INIMATG%$RESG%(RECLOOP G%*THRESHOLDG%%LISTG%'PARAMSG%"uG%'TOSUBSG%"JG%&SARGSG%"jG%"kG%"mG6"FHC.@$29#"" #-%&ERRORG6#%;wrong~number~of~parametersG>8'-T#6&7#&9"6#;"""FM.8%.8&.8(@%2FMFLC )>8@<#&FY6#;""$FL@%-%'memberG6$.%)rememberGFaoC$>8.7#F[p>Fao-%&minusG6$Fao<#F[p >F_p7">8;-Fio6$.%%listGFao@$Fip>Fao-Fcp6$Fao<#F\q>8<-%#opG6#-%'selectG6%%%typeG Fao/-%*identicalG6#.%'paramsG-F]q6#%%nameG@%0Fdq%%NULLGC%>Fao-Fcp6$Fao<#Fdq>8>- %$seqG6$/-Ffq6$8--Ffq6$FMFdq(%"bGFes/Fes;Ffn-%%nopsG6#Ffs>Fdq-F`s6$FhsFjs>F^sFg r@$0Fao<"-FO6$%2invalid~argumentsG-Ffq6#FaoC&>F_pFgp>Fip%&falseG>FdqFgr>F^sFgr> 8/,&-F]t6#FSFfn!"#Ffn>F\o-%%subsG6$7#F^s-T%6'FSFhnFjnF\oF^u>80-F]t6#F\o>FS-Fiu6 $7$/Fjn,&FjnFfnFbu!""F^s-T'6$FSFjn>8)7$81Fdq>8*,$*&-%(convertG6%-Ffw6$7$&FS6#Ff n-F`s6$*&&FS6#,&FesFfnFMFfnFfn&8,6#FesFfn/Fes;""!,&FbuFfnFivFfn%"+G%'hornerGFjn Ffn-Ffw6%&FS6#FduF[yFjnFivFiv@'30F_pFgp4Fip@%/-F]t6#F^wFfn>8$-%'subsopG6$/""%Fg r-%)procmakeG6#-%&&procG6&F^wFgpF`p-Fiu6$7$/FjnF`w-F`s6$/Fcx-&%&&argsG6#Ffu6#,( F`wFfnFbuFivFesFfnFfxFbw>Fiy-F[z6$F]z-F`z6#-Fcz6&F^wFgpF`p-Fiu6$F\o-%$&ifG6$-F` s6$-Ffq6#7$/F`wFes-FhnFex/Fes;Fhx,&F`vFfnFivFfn-Fiu6$7$Fhz-F`s6$/Fcx-F][l6$Fa[l FdqFfxFbwFipC$>89-%)&statseqG6%-Ffq6#-Fiu6$F\o7#-F`s6$-%$&:=G6$&8=FexFd\lFe\l-% %&forG6(85F`vFfnF`w%%trueG-F^^l6$&Fa^l6#Fe^l-Fiu6$7$/FjnFe^l-F`s6$/Fcx&Fa^l6#,( Fe^lFfnFbuFivFesFfnFfxFbw7#-.F`s6$Fi^l/Fe^l;FhxF`w@%Fey>Fiy-F`z6#-Fcz6&F^w7$Fe^ lFa^lF_pFb]l>Fiy-F`z6#-Fcz6&F^wFa`lF_p-Fiu6$F\o-F\\l6$-F`s6$-Ffq6#7$Fc\l7#-F`s6 $-Fhn6#8?/Feal;FhxFesFe\lFb]lC%>Fb]l-Fd]l6%-Ffq6#-Fiu6$F\o7#-F`s6$-F^^l6$-%&eva lnG6#(Fa^lFes-Fhn6#,(F`vFfnFbuFivFesFfnFfx-Fc^l6(Fe^lF`vFfn,&F`wFfnFivFfnFf^l-F d]l6$-F^^l6$-Ffbl6#(Fa^lFbu-Fiu6$7$F^_l-F`s6$/FcxFeblFfxFbw-F`s6$-F^^l6$Febl-Ff bl6#(Fa^l,&FesFfnFfnFfnFfx-Fiu6$7$FhzFiclFbw@$2FfnFfy>Fb]l-Fiu6$F\o-F\\l6$F^\lF b]l@%-%$hasG6$-F[z6$/FfnFgrFSFjn>Fiy-F`z6#-Fcz6&F^w7$Fe^l-F`s6$Febl/Fes;FhxFbuF gpFb]lC)@$0F[xFhx-%'RETURNG6#-9!6$-T)6$9$9%F\gl>8:-%&roundG6#-%&evalfG6#,$*(Fbu FM-%#lnG6#FMFiv-%)LambertWG6$Fiv,$*&FgglFfnFbuFfu#FivFMFfnFfu>877$7#-F`s6$,$*&& FS6#,&FduFfnFesFivFfnF^yFivFiv/Fes;FfnFbu-F`s6$7%-%"$G6$Fhx,&FesFfnFivFfnFfn-Fa il6$Fhx,&FbuFfnFesFiv/Fes;FfnFix>86-Fiu6$F\o7#-F`s6$-Fhn6#,&F`vFfnFesFivF[il>8+ -%&arrayG6$F\ilF\il>88-Ffjl6#F\il>Fiy-F`z6#-Fcz6&F^w7*8284Fe^l8A8B8C83F[flFgp-F \\l6%1F`wF^glFb]l-Fd]l6'-F^^l6$Fb[m-.Ffjl6%F\ilF\ilFahl-F^^l6$Fc[m-F`\m6$F\ilFj il-F^^l6$Fg[m-.Ffw6%,(F`wFfnF`vFivFfnFfn%%baseGFM-F\\l6$/&Fg[mF\xFfn-F^^l6$Fc[m -F`\m6$F\il-Ffq6$Ffo-&%'linalgG6#.%)multiplyG6$-Ffjl6#Fahl-FfjlFe\m-F\\l6%/Fg[m 7#Ffn&Fc[mF\x-Fd]l6$-Fc^l6&Fe^l-.F[z6%Fdel/-.F]t6#Fg[mFgrFg[mFf^l-Fd]l6$-F^^l6$ Fb[m-F`\m6%F\ilF\il7#-Fg_l6$7#-Fg_l6$-.%$addG6$*&&Fb[m6$Fd[mFf[mFfn&Fb[m6$Ff[mF e[mFfn/Ff[mF\il/Fe[mF\il/Fd[mF\il-F\\l6$/Fe^lFfn-F^^l6$Fc[m-F`\m6$F\il7#-Fg_l6$ -F^`m6$*&Fb`mFfn&Fc[m6#Ff[mFfnFf`mFh`m-F^`m6$*&-F^`m6$*&&Fb[m6$FfnFd[mFfn&Fb[m6 $Fd[mFe^lFfnFh`mFfn&Fc[mFj^lFfn/Fe^lF\il@$3FcyFey?&FesF\oFf^l>-Fiy6#-Ffq6#-Ffq6 $FfnFes-Ffq6$FMFes-Ffq6#FiyFHFH6*%*formatrecGc*formatrec%:-62%3goodinitvalues/r ecGc3goodinitvalues/rec%:-67%-listprimpartGc-listprimpart%:-73%/rectohomrecbisG c/rectohomrecbis&:-1036#/%+modulenameG%%gfunG M6R0 I%:-74G6" M6R0 I%:-26""%6#/%+modulenameG%%gfunG M6R0 I%:-59R6$%$polG%"nG6"F'F',&-%$maxG6$!""-%#opG6#-T#6$9$9%"""F5F5F'F'6$%)myisolve Gc)myisolve%:-88F' M6R0 I$:-2R6&%$DeqG%#eqG%%yofzG%'inipolG6A%"yG%"zG%$deqG%"PG%"uG%"dG%"iG%#d1G%"kG%"A G%"CG%#DgG%"gG%"jG%(ord_eqnG%"cG%"fG%(inhomogG%$eqnG%%inipG%%inidG%%deq0G%$iniG %#P0G%"vG%%deq1G%#P2G%'tosubsG%&lvarsG%'reduceG%"YG6"FIC@>8'-T#6&7#&9"6#;""#9#. 8$.8%87>8)-%'degreeG6$FLFX>89-%%subsG6$/FZFX-T%6&7$9$9&FWFY88>8+,&-%%nopsG6#F\o """!"#F^p@&/Fgn""!-%&ERRORG6%%Gnot~a~polynomial~in~the~right~variableG9%FX/FgnF ^pC$>8>,&*&-%&coeffG6%FLFXFbpF^p-F_q6%FLFXF^p!""FcqFZFcq@(/F[qFbp-%'RETURNG6#Fe o4-%$hasG6$F[qFZ-Fgq6#-T'6%-%$mapG6%%(collectG-F^o6$/FX,&FZF^pF[qF^pF\oFZFXFZ/F ioFbp-Fgq6#-F`r6%-F^o6$/FX,$F]qFcqF\oFXFZ>80-T)6&-%%diffGFjnFLFX.8(@$-F[r6$FdsF X-Fgq6#-T+6%Feo-%'normalG6#*&FLF^pFdsFcqFfo>8/-%$remG6%,$*(F[tF^pFdsFcq-Fis6$FL FZF^pFcqFLFX>Fds-Ffs6&&F\o6#,&FioF^pFUF^pFLFXFjs@$F]tF_t>8&-Fcr6&F[u7#-%$seqG6$ ,$*(&F\o6#8*F^pF[tF^pFdsFcqFcq/Fdv;F^p,&FioF^pF^pF^pFLFX>85-%&evalbG6#0-%#opG6$ F^pFiuFbp@%Fiv>82*&FgnF^pFgvF^p>Fcw*&FgnF^pFioF^p>8?/-Fis6$-846#FX7#-%"$G6$FXFi o-F`r6%FiuF]xFX>8@7#-F^v6$-Fis6$F\x7#-Fax6$FX8,/F_y;Fbp,&FioF^pFcqF^p>&866#FbpF \x>8A-F^o6$7$/%#_PGFL/%#_YGFXR6#%"QGFIFIFI-F[u6%FeoF]zF_zFIFIFI?(F_yF^pF^pFcw%% trueG>&Fey6#F_y-Fer6&,&-Fis6$&Fey6#,&F_yF^pFcqF^pFZF^p*&-F^o6$Fhw-Fis6$F_[lFXF^ pFitF^pF^pFfx.%,distributedGFhy>Fhw7#-F^v6$/Fjx)8B,&F_yF^pF^pF^pF`y?(F_yFbpF^pF cwFfz>Fhz-F^o6$FhwFhz>8--%&arrayG6%;F^p,&FcwF^pF^pF^p;F^pFcw%'sparseG>8.-Fh\l6$ Fj\lF]]l>&Ff\l6$F^pF^pF^p?(F_yF^pF^pF[]lFfz?(FdvFbpF^pFbyFfzC$>83-Fer6$-F_q6%F_ [lF_\l,&FdvF^pF^pF^pFX?(81FbpF^p,&FgnF^pFcqF^pFfz>&Ff\l6$F_y,(*&F`^lF^pFioF^pF^ pFdvF^pF^pF^p-F_q6%Fi]lFXF`^l@$Fiv?(F_yF^pF^pF[]lFfzC%>Fi]l-F_q6%F_[lF_\lFbp?(F `^lF^pF^pFa^lFfz>&Ff\l6$F_y,&FfwF^pF`^lF^p-Fet6#Fg^l>&F_]lFiz-F_q6%Fi]lFXFbp>Fi u-F^o6$7#-F^v6$/&88 =-Fbo6#7$FiuFe`l>Fcw-T16$-T/6%FablFZFZFZ@%/FcwFbp>8:<"C*>F_clF`cl?(FdvFbpF^pFI- F[r6$Fen---%#@@G6$%"DGFdvF^xFfyFI>8;-%(convertG6$-F^o6$Fen7#-F^v6$*(---Ficl6$F[ dlF`^lF^xFfyF^p-%*factorialG6#F`^lFcq)FZF`^lF^p/F`^l;Fbp,&FdvF^pFcqF^p%"+G@%2Fd vFcwC%>FL-F^o6$/FX,&F]dlF^p*&)FZFdvF^pFXF^pF^pFL>Fen-T36'FLFZFX,&FcwF^pFdvFcqFf z@'/-F\p6#FenFbp@%2FbpFdv>FenF]dl>Fen%)infinityG2F^pFdfl>FenF]dlC$>Fen,&F]dlF^p *&F\flF^p&Fen6#F^pF^pF^p>Fen-F^o6$-F^v6$/Fdv-%"OG6#)FZ-%%ceilG6#-F_w6$FU-F_wFcv /Fdv-%'indetsG6$Fen-%)specfuncG6$)-%*identicalGFf`l%)rationalGFjglFen>FenF]dl>F ]dl-%%evalG6#-F^o6$7$/FjglFbp/FZFbpFen@'/F]dlFbp>Fgo-T56'-F^o6$/FXFZF\oFXFZFgo, &FcwF^pFcqF^p/F]dlFjfl>FgoF`cl>Fgo-F\jl6'-Fcr6%Fer-F^o6$/FX,&F]dlF^pFZF^pF\oFZF XFZF`clFajl@$30FgoF`cl0FenFjflC$>F_cl-%'seriesG6%-F^o6$/Fb`l,&FenF^pF]dlFcq-F_d l6$-F^o6$Fgo7$-F^v6$*(FgdlF^p)Fb`lF`^lF^pF[elFcq/F`^l;FbpFajl-Fjgl6#)Fb`lFcwFbe lFZFjfl@$4-F[r6$F_cl-FjglFbgl>F_cl-F_dl6$F_cl%(polynomG@&-%%typeG6$F_clFd[m>F_c l-F\jl6&FablFXFZ<#-F^v6$/Fgdl*&-F_q6%F_clFZF`^lF^pF[elF^p/F`^l;Fbp-%$minG6$Fajl ,&-F_w6$-F\p6#F_clF_clF^pFcqF^p-Fc]mF_]m>F_cl-F\jl6&FablFXFZ<#-F^v6$F[^mFc\m@%/ F_clF`clFiu<$-F_wFh^mFiuFIFI66%,formatpoleqGc,formatpoleq%:-61%-formatdiffeqGc- formatdiffeq%:-60%+makediffeqGc+makediffeq%:-83%(mygcdexGc(mygcdex%:-87%.algebr aicsubsGc.algebraicsubs$:-2%'lindepGc'lindep%:-72%+indicialeqGc+indicialeq%:-69 %-firstnonzeroGc-firstnonzero%:-59%.algeqtoseriesGc.algeqtoseries$:-4%6goodinit values/diffeqGc6goodinitvalues/diffeq%:-666#/%+modulenameG%%gfunG M6R0 I%:-97G6" M6R0 I%:-49R6'%$PolG%"xG%"yG%$ordG%$optG69%$polG%"aG%"uG%"iG%"jG%$ptsG%&alphaG%%mini G%$degG%&thetaG%%jminG%#slG%'lastptG%"pG%#puG%$nb2G%$resG%"rG%#nbG%"qG%#a0G%+no rmalizerG%#eqG6#%RCopyright~Bruno~Salvy,~INRIA~Rocquencourt,~FranceG6"C/>%-_Env ExplicitG%&falseG>8$-%(collectG6%9$7$9&9%%&evalaG>8,-%'degreeG6$FJFP>8)-%'selec tG6$R6#F&FDFDFD0&FN6#""#""!FDFDFD7#-%$seqG6$7$8'-%&coeffG6%FJFPFco/Fco;F]oFT>FY 7#-F`o6$7$&Fco6#"""-%(ldegreeG6$&FcoF[oFQ/FcoFY>86F]o>80&FYF_p?(FcoF\oF`p-%%nop sG6#FY%%trueGC(>8+%)infinityG?(8(FcoF`pF\qF_qC$>8-*&,&&&FY6#FeqF[oF`p&FipF[o!"" F`p,&&F\rF_pF`p&FipF_pF_rF_r@$2FhqFbqC$>FbqFhq>8.Feq>Fgp,&FgpF`pF`pF`p>&8*6#Fgp ,$FbqF_r>FcoFhr>Fip&FY6#Fco>83F]o>F]s<#-F`o6$&F]sFcs/Fco;F`pFgp@$/9#""&>F]s-Fen 6%%%typeGF]s%,nonnegativeG?&8/F]sF_qC(>85-%&denomG6#Fgt>81-FL6$-%%subsG6%/FQ)FQ Fjt/FP*&)FQ-%&numerGF]uF`pFPF`pFJFQ@$0-Fbp6$F_uFQF]o>F_u-FL6$*&F_uF`p)FQF^vF_rF Q>87-FL6$-Feo6%F_uFQF]oFP@$0-Fbp6$FfvFPF]o>Ffv-FL6$*&FfvF`p)FPF]wF_rFP?&8&&-%(s qrfreeGF^wF[oF_qC%>88-%*traperrorG6#-FR6#-%'RootOfG6$&FewF_pFP@'0F[x%*lasterror G>F[x7#F[x0-%#opG6$F`pFix%Nreducible~RootOf~detected.~~Substitutions~areG-%&ERR ORG6#F[x>F[x-%$mapG6%Fcu&F[xF[oFax?&F[xF[xF_q@'/&FewF[oF`pC(@%-Fdt6$F[xFbx>89FR >F`z%'normalG>&8%6#F]oF[x>82F_u?(FcoF`pF`p,&9'F`pF_rF`pF_qC%>Fhz-FL6$-Fcu6$/FP, &&Fez6#,&FcoF`pF_rF`pF`p*&FQF`pFPF`pF`pFhzFQ>8:-Feo6%FhzFQFco@%/-FV6$Fi[lFPF`p> &FezFcs-F`z6#,$*&-Feo6%Fi[lFPF]oF`p-Feo6%Fi[lFPF`pF_rF_r-F`y6#%0unforecast~case G>Fes,&FesF`pF`pF`p>&846#Fes7$-F`o6$7$Fa\l,&FgtF`p*&FcoF`pFjtF_rF`p/Fco;F]oFjz7 $-%"OGF_p,&FgtF`p*&F[[lF`pFjtF_rF`p/F[[lF]o?(FDF`pF`pFjyF_qC$>FesF^]l>F`]l7#7$F \^lFgt?&Fco-T#6'-Fcu6$7$/FQ)FQFjy/FP,&F[xF`pFg[lF`pF_uFQFPFjzF`pF_qC$>FesF^]l>F `]l7$7$F[xFgt-F`o6$7$&FeqF_p,&FgtF`p*(,&&FeqF[oF`pF`pF`pF`pFjyF_rFjtF_rF`p/FeqF co7#-F`o6$&Fa]lFcs/Fco;F`pFesFDFD6$%3algeqtoseries/doitGc3algeqtoseries/doit%:- 49FD M6R0 I%:-82R6$%"lG%"xG6-%"aG%#a0G%"kG%$eqnG%"uG%"vG%"wG%$denG%"iG%"zG%"cG6"F3C7>8$9$ ?(8&"""F:F3/-%#opG6$F:F6""!>F6-%'subsopG6$/F:%%NULLGF6>8%F<>F9,&F9F:!""F:@$2-%% nopsG6#F6""&-%'RETURNG6#%%FAILG>F67#-%$seqG6$*&-F=6$8,F6F:FGFJ/Fgn;""#FM>8'-%'n ormalG6#,,*&,.*(&F66#""%F:&F66#F:Fjn&F66#FjnF:""'*&FioF:&F66#""$Fjn""**(FdoF:Fg oF:F]pF:F[p*&F]pFjnFgoFjn!"'*(FioFjnF]pF:FgoF:F:*&FdoF:FioFjn!#;F:9%FjnF:*&,.Fe p!#KFdpFPFapF[pFco"#=F\p"#FFbp!#CF:FgpF:F:FdpF[pFbp!#=Fco"#7@'/F\oF?>8)F:/-%'de greeG6$F\oFgpF?FQ>Fcq-F=6$F:7#-%&solveGFgq>8+-F^o6#,0*&FioFjnFcqFjnFfo*(F]pF:Fc qFjnFgoF:!"$*(FioF:FcqFjnFgoFjnFJ*(FioF:FcqF:FgoFjnFer*&FioFjnFcqF:"")*(F]pF:Fc qF:FgoF:Fer*&FioF:FgoFjn!"#@$/F_rF?FQ>8**&-F^o6#,$*&,*FhrFjn*$FioFjnFfoFjrFer*& F]pF:FgoF:FerF:FcqF:F\sF:F_rFJ@$5-%%typeG6$F`s%'negintG/F`sF?FQ>8--F^o6#,,*(F]p F:FcqF:FgoFjnFer*(FgoF:FioFjnFcqF:F:*(F]pF:FcqF:FioF:F_p*&F]pF:FgoFjnFer*&FioFj nFgoF:Fjn@$/FatF?FQ>8(,$*(-F^o6#FfsF:FgoF:FatFJFJ>Fat,$*&FatF:F_rFJFjn-%)userin foG6)F_p.%%gfunG%6candidate:~hypergeom(G7$F]uFcq7#F`s*&FatF:FgpF:%")G>8.,$*F`v*.F`vF:,(F]uF:FgnF:FJF:F:,(FcqF:FgnF:FJF:F:FatF:,(F`sF:FgnF:FJF:FJ FgnFJ@$0F`vFenFQ-Ffu6(FjnFhu%=hypergeom~found,~parameters:GF[vF\vF]v-FR6#-%)sim plifyG6$*(FGF:)FgpF9F:-%*hypergeomG6%F[vF\vF]vF:FcxF3F3F3F3 M6R0 I%:-34R6&%"pG'%%ldeqG%%listG'%#lyGF(%%yofzG6B%"iG%%deqsG%$ordG%$iniG%"yG%"zG%$l coG%%varsG%"nG%"hG%"uG%'tosubsG%)origdiffG%"jG%"kG%#l2G%"lG%#yyG%'dorderG%%jmax G%%indsG%#ppG%&lindsG%"vG%$monG%%lindG%&lcoefG%%lmaxG%&basisG%%backG%#uuG%%sing G6"FMC9@$0-%%nopsG6#9%-FR6#9&-%&ERRORG6%%@not~the~same~number~of~elementsGFTFW- T#6%9'8(8)>8,FQ>8/-%'subsopG6$/""%%%NULLGRFMFMFMFM9"FMFMFM?(8$"""FjoF]o%%trueGC )>&8%6#Fio-T%6&7$&FTF`p&FWF`p.85.F[o&8'F`p>&8&F`p,&-FR6#F^pFjo!"#Fjo>8*&F^p6#F` q>8<-%%diffG6$-&FjnF`p6#F[o7#-%"$G6$F[oF]q>8;,&*&&F^p6#FjoFjoFdq!""Fir-%$addG6$ *(&F^p6#81FjoFdqFir-Fjq6$F\r7#-Far6$F[o,&F`sFjoFbqFjoFjo/F`s;""#,&F]qFjoFjoFjoF ir>-F_o6#FhqFdr>&8>F]t7#-%'coeffsG6%Fdr7#-%$seqG6$-Fjq6$F\r7#-Far6$F[oF`s/F`s;" "!,&F]qFjoFirFjo&8=F]t@%-%$hasG6$9$%"DG>89-%(convertG6$FhuFjq>F[vFhu>8.-%%subsG 6$7#-Fgt6$/-%#opG6$F`uFfpF]r/Fio;FjoF]oF[v>8:<#-Fgt6$FiqF\w>807#-Fgt6$/Fio-F_oF `p/FioF_w>F[v-%'selectG6%Ffu-%'indetsG6$Fav-%)specfuncG6$%)anythingGFjq<#-Fgt6$ F]rF\w?(8?F`uFjoFM0F[v<">F[v-F]x6%Ffu-%$mapG6$FjvF[vFjq?(FMFjoFjoFM-Ffu6$FavF_w >Fav-%%evalG6#-Fcv6$FdwFav>F[vFav>8+<#-Fgt6$FftF\w@$4-%%typeG6$Fav-%(polynomG6$ -%(ratpolyG6$FexF[oF^z@%/FUFjo-FY6&Fhu%5not~a~polynomial~in~G-FjvFV%4and~its~de rivativesG-FY6&FhuFa[lFb[l%6and~their~derivativesG>8@F^z>F_w-%&unionG6$<#FjoF_w ?(86FjoFjoFMF[pC3>Fav-%(collectG6&FjyF^z.%,distributedG%'normalG>Fav7#-Fct6%Fav F^z.84>F]]l7#F]]l>Fav-Fc\l6&-F]v6$7#-Fgt6$*&&FavF_sFjo-F_o6#&F]]lF_sFjo/F`s;Fjo -FR6#Fav%"+GF^zFe\lFg\l>FdqFi\l>83-%&minusG6$<#F]]lF_w@$/Fd^lF\y[>F]]lF_]l>8A-% &evalbG6#/-Ff^l6$Fd^lFh[lF\y@%4F^_l>Fhq-Fjv6$FjoFc_l>Fhq&Fd^lFhr-%'memberG6%Fhq F]]l.82>F\t-Fc\l6%*&,&-Fjq6$-8-F^r7#-Far6$F[o,&F_\lFjoFirFjoFjo-F]v6$7$-Fgt6$*& &FdqF_sFjoFj]lFjo/F`s;Fjo,&F``lFjoFirFjo-Fgt6$Fcal/F`s;,&F``lFjoFjoFjo-FR6#F]]l Fa^lFirFjo&Fdq6#F``lFirF^zFe\l>F_t7#-Fct6%F\tF^zFbu>Fh[l-F[\l6$Fh[l<#Fbu@$F^_l? &FioF_wF[p@$-F]`l6%Fhq7#&FcuF`pF_`lC%>F_clF^cl>Fiw-Fc\l6%-F]v6$7%-Fgt6$*&&&F`tF `pF_sFjo&F_clF_sFjoFeal*&&F\dlF`blFjoF\tFjo-Fgt6$Fjcl/F`s;F\bl-FR6#F\dlFa^lF^zF e\l>F\dl7#-Fct6%FiwF^z-%&evalnG6#F_cl>F_w-F[\l6$F_w<#Fhq>Fav-Fjq6$FavF[o>Fav-Fc v6$/Fi`lFjn-Fc\l6$-%)primpartG6#-%&numerG6#,&Ff`lFjoFavFir7#-Fgt6$-Fjq6$Fh`lF[u /F`s;F`uF]al>8C%&falseG?(FioFjoFjoF]o4Fifl@$3/FjpF\y/-Fcv6$/F[oF`uFeqF`u>FiflF[ p@%Fifl>F[qF\yC,>87Fir?(FioFjoFjoF]oF[pC&>-Fiu6#F]rF]hl>Fjp-T'6'F^pFiv-Fjv6#Ffp Fjp,(F_\lFjoFbqFjoFjxFjo>88-F`x6$Fjp&%#_CG6#Fex@$0FghlF\yC$>Fjp-Fcv6$7#-Fgt6$/& FghlF_s&F[il6#,&FiglFjoF`sFjo/F`s;Fjo-FR6#FghlFjp>Figl,&FiglFjoF]jlFjo>F[q-F[\l 6#-Fgt6$-Fcv6$FhvFjpF\w>Figl-%$maxG6%-Fgt6$-Fjv6$Fis-Fjv6$F`u-Fjv6$F`u-Fjv6$Fjo Fio/Fio-F]x6%FfuF[q%#@@G-Fgt6$Fjo/Fio-F]x6%FfuF[q<#-Fgt6$F]hlF\wF`u>8B-Fbp6#7$F av-FjnF^r>Figl-Fjjl6$Figl-T)6%Fc\mFjnF[o>&Fdr6#F`u-%*traperrorG6#-%&limitG6%F[v Fcgl%&rightG@$5/F_]m%*lasterrorG-Ffu6$F_]m%)infinityG>FiglFir?(F`sFjoFjoFiglF[p C$>F[v-F]v6$-Fjq6$F[vF[oFiu>&FdrF_s-Fcv6$FcglF[v>F[q-Fahl6&Fc\mFjnF[o-%'removeG 6%Ffu-Fcv6$F[q<#-Fgt6$/---Fi[m6$FiuF`s6#FjnF`]mFh^m/F`s;F`uFiglFfx@%/F[qF\yFav< $-Fjv6#F[qFavFMFM6*%(getnameGc(getname%:-65%-formatdiffeqGc-formatdiffeq%:-60%6 goodinitvalues/diffeqGc6goodinitvalues/diffeq%:-66%*nbinicondGc*nbinicond%:-896 #/%+modulenameG%%gfunG M6R0 I%:-67R6(%$recG%"uG%"nG%$iniG%%flagG%"pG64%#n0G%&orderG%"iG%%indsG%'minindG%'ma xindG%$sysG%"rG%$solG%"bG%"aG%(minind2G%%lminG%"jG%"kG%$rejG%&inds2G%&dorejG6"F >C5>8%,&-%%nopsG6#9$"""!"#FG>8),&FAFG!""FG@%-%%typeG6$9'%$setG>8'-%$mapG6$%#opG -%'indetsG6$FQ-9%6#%(integerGC$>FT<">FJ-%$maxG6$FJFQ@$/9#""'>FJ-F_o6$FJ9)>8$-T# 6$-%%subsG6$/9&,&FbpFGFAFL&FF6#FCFbp>FJ-F_o6%FJ-FX6#FT,&FjoFGFLFG>8(-%$minG6$Fi p""!>8/,(-F_o6%FKF[qFipFGFAFLFGFG@%2F]qFcq>807$F]qFcq>Fjq7#F]q>8+-T%6%FFFgnFbp? (81FGFG-FD6#Fjq%%trueGC'>8*<$-FX6#FQ-%$seqG6$-F_p6$/Fbp8&F_r/Fds;-FX6$FdrFjq,&F JFGFAFL@$/FjrF\o@%49(-%'RETURNG6#<#-F_s6$/-Fgn6#Fds&%#_CGFgt/Fds;F]qFJ-F`t6#<#- F_s6$Fet/Fds-%&minusG6$<#-%"$G6#F[u<#-Fgu6#;FaqFK>8.-T'6%Fjr7#-F_s6$FftFjt.8->8 ,-%*traperrorG6#-&%'linalgG6#%)linsolveG6$F^vFfv@&35/Fhv%%NULLG/Fhv%*lasterrorG /FdrFer-%&ERRORG6#%Fhv-%(convertG 6$Fhv%%listG>FT-FZ6$Fhv&%#_tG6#%)anythingG>853F^t/FdrFG>Fdr-F_o6#-FX6#-FV6$FX-F Z6$7$FFFQ&FitF\y@$/Fdr,$%)infinityGFL>FdrFL?&FdsFTFgr@%3-%'memberG6%FdsFhv.8255 54F_y2FG-FD6#-%'selectG6%%$hasGFhvFds2Fiz,&F]qFGFLFG2FA,&FizFGF]qFGC%>Fdr,&FdrF GFGFG>Fhv-F_p6$/Fds&Fit6#,&F]qFGFdrFGFhv>&83FgtFfw>Fd\lFiz>Fjr<#-F_s6$/-Fgn6#,( FdsFGF]qFGFLFG&FhvFgt/Fds-Fcu6$<#-Fgu6#;FG-FD6#Fhv<#-F_s6$Fd\l/FdsFT@%333-%(has typeG6$-%'removeG6%FOFjr/-FgnF\y%%nameGF\z2-F_o6#-FX6#-FV6%%'degreeG7#-F_s6$-FX 6$""#Fds/FdsFjr-FZ6$FjrF\z""$4-Fd[l6$Fjr%'RootOfG-FO6$Fb_l-Ffx6#-%(polynomG6$%) rationalGFi_lC)>FTFi_l>84-%%sortG6#7#-FX6#-FV6$FX-FZ6$FjrFh^l>Fjr-F_p6$7#-F_s6$ /Fft&FgnFgt/FdsF[al<#-F_s6$,&-FX6$FGFdsFGFe_lFLFh_l>Fhv-F_p6$-%&solveG6$<#-FX6# -Fe^l6%Fb^l-&%(grobnerG6#%'gbasisG6%Fjr7$Fip-F_s6$F]blF^bl.%%plexGF\z<#Fgcl7#Fg cl>FdrFL?(FdsFGFGFg]lFgr@$/F_]l&Fgn6#&F[alFgt@%3F_y/-FD6#-Fb[l6%Fd[lFhvF_]lFG>F d\lFdsC%>Fd\lFfw>FdrF[\l>Fhv-F_p6$/Fadl&Fit6#FdrFhv<#-F_s6$/-FgnFbdlF_]l/Fds-Fc u6$Fc]l<#-F_s6$Fd\l/FdsFf]lFjrF>F>6(%-firstnonzeroGc-firstnonzero%:-59%(makerec Gc(makerec%:-84%,systomatrixGc,systomatrix&:-106F> M6R0 I%:-19R6"6+%'resultG%#exG%(methodsG%'methodG%"yG%"xG%"sG%%unknG%%exprGF$F$C+@$0 &9"6#""".%(stampedG-%'RETURNG6#-T%6#-T#6$""#F3>8,&F36#FA>8+&F36#""$>8&&F36#""%> 8(-%#opG6$""!FG>8)-FS6#FG>8%FC?&8'FL%%trueGC'>8*-%*traperrorG6#-T'6&F6FenFWFgn@ $/F[o%*lasterrorG\-%)userinfoG6'FJ.%%gfunG%,Trying~the~GFgnF[o>8$-T)6%F[oFWFQ@$ 0F]p%%FAILGC$-Fgo6(FAFio.%$TheGFgn.%1seems~to~satisfyGF]p-F96#7$-T+6&F[oF]pFQFW FgnFcpF$F$6,%*typecheckGc*typecheck&:-107%-listtodiffeqGc-listtodiffeq%:-19%-li sttoseriesGc-listtoseries%:-24%(s2d/s2dGc(s2d/s2d&:-104%(inicondGc(inicond%:-70 6#/%+modulenameGFjo M6R0 I$:-9=6"%&falseGE\[lZ%'arccotGR6$%"yG%"zGF$6$%)operatorG%&arrowGF$7%,&*&,&"""F3 *$9%""#F3F3-%%diffG6$-9$6#F5F5F3F3F3F3<$^#F3^#!""<"F$F$F$%$SsiGRF)F$F,F$7%,(*&F 7F3F5F3F3-F86$F7F5F6*&F5F3-F86$FGF5F3F3<#%)infinityG<#7$""!FPF$F$F$%'AiryBiGRF) F$F,F$7%,&-F86%F:F5F5F3*&F5F3F:F3F@FLFAF$F$F$%)LommelS2GR6&F*F+'%#muG%(integerG %#nuGF$F$F$7%,**&F5F6FUF3F3FFF3*&,&F4F3*$9'F6F@F3F:F3F3)F5,&9&F3F3F3F@<$FPFMFAF $F$F$%'arccscGRF)F$F,F$7%,&*&,&F4F6F@F3F3F7F3F3*&,&*$F5""$F3F5F@F3FGF3F3<%F@FPF 3FAF$F$F$%(arcsinhGRF)F$F,F$7%,&FFF3*&F2F3FGF3F3<%F>FMF?FAF$F$F$%%coshGRF)F$F,F $7%,&F:F3FUF@FLFAF$F$F$%$erfGRF)F$F,F$7%,&FFF6FUF3FLFAF$F$F$%(KummerMGR6&F*F+Ff nFhnF$F,F$7%,(*&F5F3FUF3F3*&,&F_oF3F5F@F3F7F3F3*&FboF3F:F3F@FLFNF$F$F$%(BesselI GR6%F*F+FhnF$F,F$7%,(*&F5F6---%#@@G6$%"DGF66#F;F8$-T#6&-F86$F:7#-%"$G6$F5Fbo7#FT7#F:F:@$-%% typeG6$Fct%$setG>Fct-%#opG6#-%'selectG6%%$hasGFctF:7%FctFLFAF$F$6$%,poltodiffeq Gc,poltodiffeq%:-34%(arccschGRF)F$F,F$7%,&*&,&F4F6F3F3F3F7F3F3*&,&F\pF3F5F3F3FG F3F3<%FPF>F?FAF$F$F$%(arccothGRF)F$F,F$7%,&F3F3*&,&F4F3F@F3F3F7F3F3<$F@F3FAF$F$ F$%%sinhGRF)F$F,F$FgpF$F$F$%(StruveHGR6%F*F+'FhnFgnF$F,F$7%/F_s,$*(),$F5#F3F6Fa oF3Fhs#F@F6-%&GAMMAG6#,&FboF3F\xF3F@""%FcoFAF$F$F$%)HankelH1GRFhqF$F,F$7%,(F[rF 3FcrF3F`sF3FcoFAF$F$F$%+WhittakerMGRF_qF$F,F$7%,&FUF3*&,(#F@FbxF3*&FboF3F5F@F3* &,&#F3FbxF3F^oF@F3F5!"#F3F3F:F3F3FcoFAF$F$F$%$ChiGRF)F$F,F$7%,(FIF3FGF6FFF@FcoF AF$F$F$%$cosGRF)F$F,F$7%,&FUF3F:F3FLFAF$F$F$%*KelvinKerGR6%%"wG%"xGFhnF$F,F$7%, **&F5Fbx-F86$F:-F[u6$F5FbxF3F3*&F5F]p-F86$F:-F[u6$F5F]pF3F6*&,&F3F3FhrF6F3,&*&F 5F6-F86$F:-F[u6$F5F6F3F3FFF@F3F@*&,(*$FboFbxF3Fhr!"%*$F5FbxF3F3F:F3F3FcoFAF$F$F $%(AiryBi2GRF[tF_tF$F$C%>FctFdt@$F`u>FctFeuF\vF$F$F]v%'arcsecGRF)F$F,F$FfoF$F$F $%'arccosGRF)F$F,F$7%,&FFF3*&F^wF3FGF3F3<%F@F3FMFAF$F$F$%'arctanGRF)F$F,F$7%,&F 1F3F@F3F=FAF$F$F$%'WeberEGRFhqF$F,F$7%/F_s*&,(*&,&FboF3F5F@F3-FgyFfsF3F3FboF@F5 F@F3FhsF@FLFAF$F$F$%$expGRF)F$F,F$7%,&F7F3F:F@FLFAF$F$F$%%erfcGRF)F$F,F$F[qF$F$ F$%'arcsinGRF)F$F,F$7%,&*&FUF3,&F3F3F4F@F3F3FFF@Ff\lFAF$F$F$%(BesselYGRFhqF$F,F $FexF$F$F$%*KelvinHerGRFhqF$F,F$7%,,Fd[lF3*&,&*&F5F6FboF6FbyF4F@F3FGF3F3*&,&*&F 5F3FboF6F6F5F3F3F7F3F3*&-F86$FJF5F3F5FbxF3*&FJF3F5F]pF6FcoFAF$F$F$%*hypergeomGR 6&F*F+%%lnumG%%ldenG6'%"jG%"nGF`t%%singG%"uGF$F$C&>8&-Fet6%,&*(-8(6#,&8%F3F3F3F 3Fd`lF3-%$mulG6$,&FctF3Fe`lF3/FctF_oF3F3*&-Fb`l6#Fe`lF3-Fg`l6$Fi`l/FctFboF3F@F\ alF:@$-Fau6$F\`lFcu>F\`l-Ffu6#-Fiu6%F[vF\`lF5@'2,&-%%nopsG6#F_oF3F3F3-F]bl6#Fbo >8'Fco/F_blF[bl>Fbbl<$F3FM>FbblFL7%F\`lFbblFAF$F$6$%,rectodiffeqGc,rectodiffeq% :-39%$ShiGRF)F$F,F$7%FfyFLFNF$F$F$%)FresnelCGRF)F$F,F$7%,(*(FhsF6F5F]pF7F3F3FGF @FIF3FLFNF$F$F$%#CiGRF)F$F,F$7%FEFcoFAF$F$F$%#lnGRF)F$F,F$7%,&FFF3F@F3FcoFAF$F$ F$%(arcsechGRF)F$F,F$FfoF$F$F$%(KummerUGRF_qF$F,F$7%FaqFcoFAF$F$F$%)LommelS1GRF ZF$F$F$7%FjnFLFNF$F$F$%(arctanhGRF)F$F,F$F[wF$F$F$%(BesselJGRFhqF$F,F$7%FfxFLFi rF$F$F$%*KelvinBeiGRF]zF$F,F$7%FazFLFirF$F$F$%*KelvinBerGRF]zF$F,F$FidlF$F$F$%% erfiGR6$F*%"tGF$F,F$7%,&FFFbyFGF3FLFAF$F$F$%'AiryAiGRF)F$F,F$FSF$F$F$%+Whittake rWGRF_qF$F,F$FixF$F$F$%&dilogGRF)F$F,F$7%,(FFF3*&,&F5F@F4F3F3FGF3F3F3F3Fco<$FPF 3F$F$F$%(arccoshGRF)F$F,F$Fc\lF$F$F$FesRF)F$F,F$FiyF$F$F$%#SiGRF)F$F,F$FDF$F$F$ %*KelvinHeiGRFhqF$F,F$Fd^lF$F$F$%*KelvinKeiGRF]zF$F,F$F`zF$F$F$%&erfc2GRFhqF$F, F$7%,(FGF3FFF6FeqFbyFLFAF$F$F$%(StruveLGRFdwF$F,F$7%/,(F[oF3FFF3FfrF@FhwFcoFAF$ F$F$%)HankelH2GRFhqF$F,F$FexF$F$F$%)FresnelSGRF)F$F,F$F`clF$F$F$%(BesselKGRFhqF $F,F$7%FjqFcoFAF$F$F$6#/%+modulenameG%%gfunG M6R0 I&:-105R6%%"sG%"xG%"yG6)%&serieG%"iG%%bigOG%%lpolG%"jG%$nbzG%+tryfactorsG6"F0C( @%0-%#opG6$,&-%%nopsG6#9$"""!""F8&-F56$F8F;>FB,&FCF8)-F56$"" #F;>8*-%%typeG6$-%%subsG6$/F>""!7#-F5F:-%%listG6#%)rationalG>8$-%$mapG6&%'serie sG7#-%$seqG6$)F;8%/Fao;FU,&T#FFgn7$-F56#Fgn-F[o 6%*&&Fgn6#-F9FfpF8'-T)6(FgnFfo-%$minG6$,&FBF8+-T# 6%9%9&9'>8,-F86%9$F;F<@%2F<""*C%>&8*6#""!-%&coeffG6%FAF;FJ?(8)"""FP-%'degreeG6$ FAF;%%trueG>&FH6#FO-%(collectG6$*&-FL6%FAF;FOFP)-F86%F:F;,(F8$-%&isqrtG6#-%&truncG6#,$*&F< FP-%'lengthG6#FK$!")>8%-F86%F6F;F[p>8&FP>8',&F6FPF[qF\o>8(-T%6&F>F[qF ;F<>FGFdq?(FOFPFP,&-%%iquoG6$FFdq-T'6&FdqF[qF;,&FF_q-F8 6%-FY6$*&F_qFPFaqFPF;F;F<>FV-FY6$*(FdqFPF_qFP-%*factorialGFWF\oF;-F86%-F`o6$7#- Fdo6$FV/FO;FJFjqFhoF;F9&FG>9%FF@%-%%typeG6$-F@6$FJF9%$setGC'>8$-%-selectremoveG6%%$h asGFUFG@$2FJ-F76#&FZ6#FJ-F<6$%>invalid~differential~equationGF?@$/F[o""!-F<6#%U the~unknown~variable~does~not~appear~in~the~equationG@$2""$FK>9'&FZ6#F:>FZ-F@F\ oC$>FZFU@$Fio>F\p<"@$3Fio-Fhn6$F\p%"DG>-Fjp6#FPF\q@$-FS6$FZ%"=G>FZ,&-F@6$FJFZFJ -F@6$F:FZ!""@$-Fhn6$FZFjp>FZ-%(convertG6$FZ%%diffG@$0-%'indetsG6$FZ-%)specfuncG 6$%)anythingGFF<#-FF6#FGF_o>FZ-%'expandG6#-%&numerG6#-%'normalG6#FZ>8*-%'select G6%Fhn-Fdr6$FZ-Fgr6$FirF`rFF?(8(FdoFJF20FhsFep>Fhs-%&minusG6$-%%subsG6$/-F`r6$F [sFGF[sFhsFjr@$4-FS6$FZ-%'linearG6#7$F[s-%$seqG6$-F`r6$F[s-%"$G6$FG8'/F]v;FJFat -F<6%FZ%Iis~not~a~linear~differential~equation~inGF[s>FZ-Fht6$7$/F[s8+-Ffu6$/-F `r6$F[s-F[v6$FG8))Fhv,&F`wFJFJFJ/F`wF_vFZ>FZ7$-Fht6$/FhvFdoFZ-Ffu6$-%&coeffG6%F ZFhvFbw/F`w;FdoFat@$4-FS6$FZ-%%listG6#-%(polynomG6$FirFG-F<6$%8,.%)rationalG>8-%&falseG@&/9#""%C%>F<&9"6#FE@$30F<. %*algebraicG0FF@%%trueG/FD""$@%5/&FI 6#FYF=/FgnFN>FF@FW>8*-T%6&9$F9F:F<@$F@C&>8+,&-%'degreeG6$F]oF9"""-Fho6$-%' lcoeffGFioF:Fjo>8'-%'seriesG6%FaoF:,&FeoFjoFYFjo>8)-%#opG6$""#F`p@%43-%%typeG6$ F`pFbp-F_q6$Ffp%*nonnegintG>9&<"C$?(8&FjoFjoF3FWC$@%-%$hasG6$F`p.%"OG>8(-Fhp6$- %%nopsG6#F`pF`p>Fbr%)infinityG@%1Fbr,(FfpFjoFeoFjo!""FjoC$>F`p-Fbp6%FaoF:*&FiqF jo,(FfpFjoFeoFjoFYFjoFjo@$4F^qC$>FeqFfq[Fhs@$0FeqFfq>Feq<#-%$seqG6$/---%#@@G6$% "DGFiq6#F96#""!*&-%*factorialG6#FiqFjo-%&coeffG6%F`pF:FiqFjo/Fiq;FfpF\sF]oF3F36 &%(getnameGc(getname%:-65%/algfuntoalgeq2Gc/algfuntoalgeq2%:-516#/%+modulenameG %%gfunG M6R0 I%:-98R6(%"rG%"uG%"nG%"fG%"zG%)inicondsG62%&orderG%'diffeqG%"PG%"kG%"pG%"aG%#rr G%$iniG%"iG%#k0G%%indsG%$resG%"lG%"cG%#aaG%"jG6"F8$-%$maxG6#-%#opG6#-%$map G6%%'degreeGFB9&>8%-%&arrayG6$%'sparseG;!""FN>8-,&-%%nopsG6#FB"""!"#Fbo>-%"DG6# FCFeo?(8'""!FboF]o%%trueGC$>8&-FS6$,&FioFbo""#FboFB?(8(FjoFbo-FX6$F^pFYF[pC&>8) -%%subsG6$/FY,&FdpFboFioF[oF^p>F^p-%$quoG6%,&F^pFboFipF[o,(FYFboFioFboFdpF[oFY> &Fen6#Fdp,&FfqFbo*&FipFbo)FJ,(F]oFboFdpFboFioF[oFboFbo@$/9#""'>&Fen6#F[o,&FarFb o-%(collectG6$*(FipFboFjqFbo-%%diffG6$-%$addG6$*(---%#@@G6$Ffo8,Fgo6#FjoFbo)FJF dsFbo-%*factorialG6#FdsF[o/Fds;Fdp,&FioFboF[oFbo7#-%"$G6$FJFdpFboFJF[o>F^p-FS6$ FboFB?(FdpFjoFboFepF[pC$>&8*Fgq-F[q6$/FY,&FdpF[oF[oFboF^p>F^p-Faq6%,&F^pFboFgtF [o*&,(FYFboFdpFboFboFboFbo,&FdpFboFboFboF[oFY>Fen7$,&*&FarFbo),&FboFboFJF[oFdpF boFbo*&)FJF]oFbo-%(convertG6$7#-%$seqG6$*&&Fht6#FioFbo)Fiu,(FdpFboFioF[oF[oFboF bo/Fio;Fjo,&FdpFboF[oFbo%"+GFboFbo-Fav6$*&FhuFbo&FenFevFbo/Fio;FjoFN@$F]rC'>8.- FV6$FS-%'indetsG6$9)-9%6#%)anythingG>8+-%&solveG6$-F[q6$7#-Fav6$/-F]xFis*&F_sFb oFgsF[o/FdsFewF[x<#-Fav6$F_sF]y>Fen-F[q6$FaxFen>Few-FP6#-FS6#Few@$30Few,$%)infi nityGF[o1FNFew>Fen-%'subsopG6$/Fbo,&-Fer6$&Fen6#FboFJFbo-F]v6$-%'seriesG6%-%%ev alG6#-F[q6%/-FC6#FJ-F]v6$7$-Fav6$F^s/Fds;FjoFew*&-%"OGFhzFbo)FJ,&FewFboFboFboFb oF[wFax-T#6%FenFCFJFJF]z%(polynomGF[oFen@$-F@6$FgzFCC*>80-%'selectG6%F@-Fiw6$Fg z%)functionGFC@$4-%%typeG6$Fgz-%'linearG6#F[]l-FE6#%;invalid~inhomogeneous~part G>Fen-F[q6$<#-Fav6$/&F[]lFis&82Fis/Fds;Fbo-F`oFi]lFen>F[]l7#-Fav6$Fe^lFg^l>Fen- Faz6$/Fbo-Fer6$FgzF[]lFen?&FdsF[]lF[p@$-F@6$FgzFdsC&>81-%&coeffG6%FgzFdsFbo>8/, $-Fir6$Fj_lFJF[o>Fen7&,&*&F_`lFboFgzFboFbo*&Fj_lFbo-FirFfzFboFbo,&*&F_`lFbo&Fen 6#FbpFboFbo*&Fj_lFbo-Fir6$F[alFJFboFbo-Fav6$,(*&F_`lFbo&Fen6#83FboFbo*&Fj_lFbo- Fir6$FdalFJFboFbo*&Fj_lFbo&Fen6#,&FfalFboF[oFboFboFbo/Ffal;""$-F`o6#Fen*&Fj_lFb o&Fen6#FablFbo>Fen-Faz6$/Fbo-Fer6%FgzF[]l%'normalGFen@$-F@6$FenFf^l-FE6#%9some~ assertion~was~wrongG>Fen-FV6%FerFenFJ@%F]r>Fax-T%6&FenFCFJFax>Fax<">F_`l-Fc\l6% FbyFCFJ@%0FaxF\dl<$-FS6#FaxF_`lF_`lF8$-%(convertG6$7$-%$seqG6$*&&9$6# ,&8%"""""#F=F=-%%diffG6$-9%6#9&7#-%"$G6$FEF8&FO>8' -%'subsopG6$/""%%%NULLGRFKFKFKFK9"FKFKFK?(8("""FhoF[o%%trueGC)>&8)6#Fgo-T%6&7$& FRF^p&FUF^p.8*.Fin&88F^p>&8;F^p,&-FP6#F\pFho!"#Fho>89&F\p6#F^q>8:-&FhnF^p6#,&Fi nFhoF[qFho@%2""!F[q>8+-%(convertG6$-%$mapG6$%'normalG7$,$*&&F\p6#FhoFhoFbq!""F\ s-%$seqG6$,$*(&F\p6#8.FhoFbqF\s-Fhq6#,(FinFhoFdsFhoF`qFhoFhoF\s/Fds;""#,&F[qFho FhoFho%"+G>F_r,$-Ffr6#FirF\s>-F]o6#FfqF_r>&8,Fct7#-%'coeffsG6%F_r7#-F^s6$-Fhq6# ,&FinFhoFdsFho/Fds;F]r,&F[qFhoF\sFho&8-Fct>8=-%%subsG6$7#-F^s6$/-%#opG6$F]rFdpF hq/Fgo;FhoF[o9$>81Fgu>8?F]r?(FgoFhoFhoF[oFioC$>84-T'6%FfvFhqFin@$0F\w,&%)infini tyGF\sFinF\sC$>Fhv-%$maxG6$FhvF\w?(FdsF\wF\sF[qFio>Ffv-Fiu6$/F^u-Fiu6$/Fin,(Fin FhoFdsFhoF[qF\s-F]o6#FgqFfv>8/<#-F^s6$F\uFbv@$4-%%typeG6$Ffv-%(polynomG6$-%(rat polyG6$%)anythingGFinFex-FW6$%1Not~a~polynomialGFdv>8@Ffx>80<$Fho-F^s6$FgqFbv?( 82FhoFhoFKFioC3>Ffv-%(collectG6%-Fiu6$7#-F^s6$/FgqFbxFbvFfvFex.%,distributedG>F fv7#-Fit6%FfvFex.83>Fc[l7#Fc[l>Ffv-Fdz6&-Far6$7#-F^s6$*&&FfvFcsFho-F]o6#&Fc[lFc sFho/Fds;Fho-FP6#FfvF\tFexF\[lFfr>FbqF_[l>F\w-%&minusG6$<#Fc[lF[z@$/F\w<"[>Fc[l Fe[l>8A-%&evalbG6#/-Fj\l6$F\wFiyF_]l@%4Fc]l>Ffq-F`v6$FhoFh]l>Ffq&F\wF[s-%'membe rG6%FfqFc[l.85>Fbt-Fdz6%*&,&-%"hG6#,(FinFhoF`zFhoF\sFhoFho-Far6$7$-F^s6$*&&FbqF csFhoF`\lFho/Fds;Fho,&Fe^lFhoF\sFho-F^s6$Fd_l/Fds;,&Fe^lFhoFhoFho-FP6#Fc[lF\tF\ sFho&Fbq6#Fe^lF\sFexF\[l>Fet7#-Fit6%FbtFexFdu>Fiy-%&unionG6$Fiy<#Fdu@$Fc]l?&Fgo F[zFio@$-Fb^l6%Ffq7#&FeuF^pFd^lC%>FaalF`al>-F]oF^p-Fdz6%-Far6$7%-F^s6$*&&&FftF^ pFcsFho&FaalFcsFhoFf_l*&&F_blFa`lFhoFbtFho-F^s6$F]bl/Fds;F]`l-FP6#F_blF\tFexF\[ l>F_bl7#-Fit6%FealFex-%&evalnG6#Faal>F[z-Fh`l6$F[z<#Ffq>Ffv-Fiu6$/Fin,&FinFhoFh oFhoFfv>Ffv-Fiu6$/F\_lFhn-Fdz6$-%)primpartG6#-%&numerG6#,&F[_lFhoFfvF\s7#-F^s6$ -F\_lF_u/Fds;F]r,&F`zFhoF\sFho>8<-Fgw6$,&F`zFhoF`qFho-T)6$-Fiu6$/Fin,(FinFhoF`z F\sFjsFho-%&coeffG6%Ffv-FhnF]_lFhoFin>86F\s?(FgoFhoFhoF[oFioC%>Fhp-T+6(F\pF_v-F `v6#FdpFhpFioF^el>87-%'indetsG6$Fhp&%#_CG6#Fdy@$0FhflF_]lC$>Fhp-Fiu6$7#-F^s6$/& FhflFcs&F]gl6#,&F^flFhoFdsFho/Fds;Fho-FP6#FhflFhp>F^fl,&F^flFhoF_hlFho>8>-Fgw6$ -F^s6$-F`v6#-F`v6$FhoFgo/Fgo-F^s6$-F`v6#FhpFbv,&FhvFhoF^elFho@$2F^elFdhlC$>F^fl F\s?(FgoFhoFhoF[oFioC&>Fhp-Fcfl6(F\pF_vFeflFhpFioFdhl>FhflFifl>FhpFcgl>F^flFbhl >Fip-Fh`l6#-F^s6$-Fiu6$F^vFhpFbv>Fip-Fcfl6'-F`p6%7$Ffv-Fhn6#Fin.FhnFgpFhnFin-%' removeG6%%$hasG-Fiu6$Fip<#-F^s6$/-FhnF^p-Fiu6$/FinFgoFgu/Fgo<#-F^s6$Fihl/FgoFip <#-F^s6$FhqFbvFio@%/FipF_]lFfv<$-F`v6#FipFfvFKFK6,%(getnameGc(getname%:-65%*for matrecGc*formatrec%:-62%)maxindexGc)maxindex%:-85%-firstnonzeroGc-firstnonzero% :-59%3goodinitvalues/recGc3goodinitvalues/rec%:-676#/%+modulenameG%%gfunG M6R0 I%:-68R6(%+listseriesG%"xG%$ordG%*nbnonzeroG%+tryfactorsG%)ordcheckG6&%%lpolG%" kG%"jG%&checkG6"F0C(>8$-%$mapG6%%(collectG-T#6%9$9%9&F<@$/F3%%FAILG-%'RETURNG6# F@@$29'-%%nopsG6#-%%subsG6$/""!%%NULLG-F56%%'coeffsGF3FF3-F56$%(fact orsGF3@$2FF,&-FH6#-FK6$FM7#-%$seqG6$-FR6$-%#opG6$"""8%FF3-F56%%(convertG7#-F^o6$7$&Ffo6 #Feo-F^o6$)&8&F_q&Fdq6#F^p/Fdq&FfoFfqF_p%"*G>8'-F56$%'normalG-%'seriesG6%-Fip6$ 7#-F^o6$*&&F36#FdqFeo&F;FirFeo/Fdq;Feo-FH6#F;%"+GF<9)@$5/F[rFN/-Fco6$FeoF[r-%"O GF_q-FB6#F3F@F0F06$%&pade2Gc&pade2%:-33F0 M6R0 I%:-20R6"6(%'resultG%(methodsG%'methodG%"sG%%unknG%%exprGF$F$C(@$0&9"6#""".%(st ampedG-%'RETURNG6#-T%6#-T#6$""$F0>8)&F06#""#>8(&F06#F>>8%&F06#""%?&8&FI%%trueGC '>8'-%*traperrorG6#-T'6%F3F@FN@$/FR%*lasterrorG\-%)userinfoG6'F>.%%gfunG%,Tryin g~the~GFNFR>8$-T)6$FRFE@$0F^o%%FAILGC$-Fhn6(FCFjn.%$TheGFN.%1seems~to~satisfyGF ^o-F66#7$F^oFNFdoF$F$6*%*typecheckGc*typecheck&:-107%0listtohypergeomGc0listtoh ypergeom%:-20%+listtolistGc+listtolist%:-21%(l2h/l2hGc(l2h/l2h%:-826#/%+modulen ameGF[o M6R0 I$:-8R6%%$eq1G%$eq2G%%yofzG6&%#y1G%#y2G%"yG%"zG6"F-C$-T#6%9&8&8'-T%6&*&-8$6#F4" ""-8%F;F<7$-%%subsG6$/F3F:9$-FA6$/F3F>9%7$F9F=F2F-F-6&%(getnameGc(getname%:-65% ,poltodiffeqGc,poltodiffeq%:-346#/%+modulenameG%%gfunG M6R0 I&:-106R6%%$sysG%"VG%"BG6+%"aG%"iG%"jG%$eqnG%%zeroG%"cG%"tG%$indG%$lcoG6"F2@%5/ 9$<"/F67"C$>9&-%&arrayG6$;"""FA7#""!-F>6%F@;FA-%%nopsG6#9%7#7#-%"$G6$FCFGC(>8(7 #-%$seqG6$/8%FC/FXFJ>8$-F>6%;FA-FH6#F6FF%'sparseG>F<-F>6$FhnF[o?(FXFAFAFG%%true G>&8+6#&FJ6#FXFX?(FXFAFAFinF`oC'>8'-%#opG6$FXF6@$-%%typeG6$Fjo%"=G>Fjo,&-F\p6$F AFjoFA-F\p6$""#Fjo!"">8,7#-%'coeffsG6%FjoFJ.8*>Fbq7#Fbq?(8&FAFA-FH6#FbqF`o@%0&F bq6#FfqFA>&Fen6$FX&Fco6#F[r&F\qF\r>&F8%-%'degreeG6$9$9&>8$"""?(F. F9F9F.2F8F1>F8,$F8""#>&8&6#F9-T#6%9%F69'>8'F9?(F.F9F9F.2FIF8C$>&FA6#,$FIF>-FD6% -%(collectG6$*$&FA6#FIF>F6F6FG>FIFP-T%6'F5F8FAF6FGF.F.6&%,powtruncateGc,powtrun cate%:-95%5powcomposesimpledoitGc5powcomposesimpledoit%:-92F. M6R0 I%:-43R6"F$F$F$@%0&9"6#""".%(stampedG-T%6#-T#6$""'F(-T)6&F+-T'6%F+&F(6#""#.%$og fG&F(6#""$&F(6#""%F$F$6*%*typecheckGc*typecheck&:-107%/seriestodiffeqGc/seriest odiffeq%:-43%-seriestolistGc-seriestolist%:-45%-listtodiffeqGc-listtodiffeq%:-1 96#/%+modulenameG%%gfunG M6R0 I%:-76G6" M6R0 I%:-28""$6#/%+modulenameG%%gfunG M6R0 I%:-61R6&%"lG%"yG%"zG%)inicondsG6%%"YG%"ZG%"PG6"F-C)@$52-%%nopsG6#9$""#2""%F2-% &ERRORG6$%Gformatpoleq:~wrong~number~of~argumentsG-%#opGF4-T#6%-F>6$F6F58$8%@$2 """9#C$>9%FD>9&FE@%-%%typeG6$-F>6$FHF5%"=G>8&,&-F>6$FHFSFH-F>6$F6FS!"">FWFS@$4- FQ6$FW-%(polynomG6$%)anythingG7$FDFE-F:6$%1invalid~argumentGFW@$/FIF8@%2F6F2@%- FQ6$-F>6$""$F5%$setG>9'F\p-F:6$FdoF\p>Fap<"-%(collectG6$-%)primpartG6$FWFDFDF-F -6$%(getnameGc(getname%:-65F- M6R0 I%:-99R6%%"lG%&orderG%$auxG6#%$resG6"F*C%>8$-%#opG6$"""-%%sortG6$9$9%@$/9#""$-% 'memberG6%F-F59&F-F*F*F*F* M6R0 I%:-84R6&%$recG%"uG%"nG%$iniG6$%"rG%"iG6"F,C$>8$-%(convertG6$7$-%$seqG6$*&&9$6# ,&8%"""""#F=F=-9%6#,&9&F=F8--%'indetsG6#9$@&-%$hasG6$F?9%-%&ERRORG6%%:expression~cannot~c ontainGFHFC4-%(convertG6$-%$mapG6%%%typeGF?<$-%'algfunG6#%)anythingG-%'radfunGF X%$andGF<@%53/9'.%*algebraicG-FT6$FC-%(polynomG6$FY9&3/F[o.%)rationalG-FT6$FC-F ao6#Fgo,&FH"""FC!""C3>8%-%&numerG6#F\p>8(-FA6$Fap)FY%)fractionG>8)-FA6$Fap%'Roo tOfG@$FjnC$>Ffp-%'selectG6%FFFfpFco>F\q-Fdq6%FFF\qFco>8*-%%map2G6%%#opGF]pFfp@$ 0-%%nopsG6#Fjq-Fbr6#Ffp?&8.FjqF`rC$>8/-Fdq6%R6$F9F4F<6$%)operatorG%&arrowGF<-%& evalbG6#/-F^r6$F]pFCFHF&816#Fgr7#-%$seqG6$-F^r6$"" #83/FhtFjr>80-%%ilcmG6#-Fct6$-F^r6$FgtFetFit>Ffp-%&unionG6$-%&minusG6$FfpFjr<#) Fgr*$F[uF^p>8+R6$%"aGF4F8&-%%sortG6$7$- F^rFer-F^r6#F\qF^v>85-Fbr6#Fjv>Fjv-%&tableGFew?(8,F]pF]pFcw%%trueG@%5-FT6$&Fjv6 #FjwF_q4-%)assignedG6#&F_t6#-F^r6$F]pF`x>&82Fax/F`x&8$Fax>F[y-Fct6$/)Fhx84)F^y* &FeyF]p-F^r6$Fgt-F^r6$FgtF`xF]p/FeyFfx>Fap-%%subsG6$7#-Fct6$F[y/Fjw;F]pFcwFap>8 '-Fhw6#7#-Fct6$-T#6$F`xF^yFdz?(FjwFcwF^pFgtF[xC$>Fgz-F_z6$F]y-%%evalG6#Fgz>Fjv- F_z6$F]y-Ff[lFew?(FjwF]pF]pFcwF[x>Fap-%*resultantG6%Fap&FgzFaxF^y>Fap-Fcp6#Fap@ %-FT6$FapFjo-%$quoG6%Fap-%$gcdG6$Fap-%%diffG6$FapFHFH-Fi\l6%Fap-%&gcdexG6%FapF^ ]lFHFHF8$-%(convertG6&9$.%,fullparfracG9%.%(sqrfreeG @%-%%typeG6$F-%"+G>F-7#-%#opG6#F->F-7#F--F/6$-%$mapG6&T#F-F49&F;F*F*6$%4ratpoly tocoeff/elmtGc4ratpolytocoeff/elmt&:-1016#/%+modulenameG%%gfunG M6R0 I%:-69R6&%$deqG%"zG%&alphaG%$valG6(%%ldegG%"iG%"vG%$resG%"jG%#ddG6"F0C(>8)-%$ma pG6$R6#%"xGF0F0F0@%/9$""!"""F8$7#-%$seqG6$,(-%(ldegreeG6$&F36#8%9%F>F K!""""#F>/FK;FN-%%nopsG6#F<>8&-%$minG6#-%#opG6#F@@$/9#""%>9'FU?(FKF>F>-FRFen%%t rueG@%/&F@FJFU>&8'FJ*&-%'tcoeffG6$&F<6#,&FKF>F>F>FLF>-%(convertG6$7#-FC6$,&9&F> 8(FM/Fdp;F=,&FKF>!"#F>%"*GF>>FcoF=-%(collectG6$-F]p6$7#-FC6$Fco/FK;F>F]o%"+GFcp F0F0F0F0 M6R0 I%:-21R6"6#%"xGF$F$@'0&9"6#""".%(stampedG-T%6#-T#6$""*F*/&F*6#""$.%$ogfG&F*6#"" #-T)6%F--T'6&F-F<8$F7F:F$F$6*%*typecheckGc*typecheck&:-107%+listtolistGc+listto list%:-21%-listtoseriesGc-listtoseries%:-24%-seriestolistGc-seriestolist%:-456# /%+modulenameG%%gfunG M6R0 I$:-7R6%%$eq1G%$eq2G%%yofzG6&%#y1G%#y2G%"yG%"zG6"F-C$-T#6%9&8&8'-T%6&,&-8$6#F4" ""-8%F;F<7$-%%subsG6$/F3F:9$-FA6$/F3F>9%7$F9F=F2F-F-6&%(getnameGc(getname%:-65% ,poltodiffeqGc,poltodiffeq%:-346#/%+modulenameG%%gfunG M6R0 I&:-107R6#%"nG6#%"iG6"F(@7/9$"""@+32""#9#-%%typeG6$7#&9"6#;F0""$7$%%listG%%name G@)/F1F:-%'RETURNG6%.%(stampedGF6-%#opG6$F,T#3/F1""%-F36$&T%6#&F76#FK%*procedur eG-FA6$FC&F76#;F0FK2FKF1-%&ERRORG6#%3too~many~argumentsG-Fen6$%1invalid~argumen tGFQ2F1F:-Fen6#%3too~few~parametersG4-F36$&F76#F0F<-Fen6$%+not~a~listGFbo4-F36$ &F76#F:F=-Fen6$%+not~a~nameGFjo-Fen6#%2invalid~argumentsG/F+F0@+3F/-F36$F57$F<- %)functionG6#F=@)F?-FA6%FCF6FH33FJ-F36$FQF<-F36$7#-%$seqG6$&FO6#8$/FjqFQ-F<6#FS FTFYFZFhnF[o-Fen6#%2too~few~argumentsGF_oFdo4-F36$FjoFhp-Fen6$%9invalid~unknown ~functionGFjoF_p/F+F:@+F.@*F?F\qF^qFTFYFZFJFhnF[oF\oF_oFdoFgoF\p-Fen6#%1invalid ~argmentsG/F+FK@-3FJ-F36$7#FV7%%'seriesGF=F=FT0F1FK-Fen6#%:wrong~number~of~argu mentsG4-F36$FboFds-Fen6$%-not~a~seriesGFboFgoF\p4-F36$FQF=-Fen6$F^pFQF_p/F+""&@ +3F?Fep-FA6$FCF60F1F:FfsF_oFdoFarFdrF_p/F+""'@+3F/-F36$F57$FdsFhp@)F?F\qF^qFTFY FZFhnF[oF^rFisF\tFarFdrF_p/F+""(@)32F,F1Fjs@)/F1F0-FA6%FCFboFH33F?-F36$FjoF<-F3 6$7#-Ffq6$Fhq/FjqFjoF\rFht2F:F1FZ-Fen6$FjnFjo/F1F,F^rFisF\tF_p/F+"")@)Ffu@)Fiu- FA6%FCFbo.%$ogfG3F?-F36$&FO6#FjoFSFhtFfvFZFgvFivF^rFisF\tF_p/F+""*@)3FguF`o@)Fi uF^wFbwFhtFfvFZFgvFivF^rF_oFdoF_p/F+"#5@(FYFZ2F1FKF^r4-F36$7#&F76#;F:FK7$FhpFhp F_p-Fen6#%2should~not~happenGF(F(6&%*optionsgfGc*optionsgf%:-32%2listtoseriesta bleGc2listtoseriestable%:-25F( M6R0 I%:-92R6'%"qG%"sG%#pkG%"xG%"nG6$%#q1G%#q2G6"F-@+/9%"")-T#6%-%(collectG6$,,-%&co effG6%9$9'""!"""*&&9&6#F?F?,(-F:6%F8 $-F36%F8%-F66$*&,&F9&FK>9%FJ@%-%%typeG6$-FD6$FNF;%$setGC'>8$-%-selectr emoveG6%%$hasGFYFK@$2FN-F96#&Fhn6#FN-F@6$%3invalid~recurrenceGFC@$/Fao<"-F@6#%1 empty~recurrenceG@$2""$FOC$>8*&Fhn6#F<@$4-FW6$Fap-Fen6#/-FJ6#%(integerG%)anythi ngG-F@6$%;invalid~initial~conditionsGFap>Fhn-FDF`oC$>FhnFY@$F]p>FapFho@$2""%FO@ %32FjqF8-FW6$-FD6$F>F;%(booleanG>9(F`r>Fdr%&falseG@$-FW6$Fhn%"=G>Fhn,&-FD6$FNFh nFN-FD6$F8(-T%6%FhnFJFK>8)-T'Ffs>Fhn-%(collectG6%Fhn7#-%$seqG6$-FJ6#,&F KFN8%FN/Fft;FcsFhs%'normalG@$4-FW6$Fhn-%'linearG6#F_t-F@6$%8Not~a~linear~recurr enceGFC@$0Fcs""!C$@$F]pC$>8+Fas?(FftFfuFNFcs%%trueG@$4-F\o6$Fap-FJ6#FftC$>F[v,& F[vFNFNFN>Fap-%&unionG6$Fap<#/Fbv&%#_CG6#F[v@$2FcsFfuC$>Fhn-%%subsG6$/FK,&FKFNF csFasFhn>Fhs,&FhsFNFcsFas@$F]p>9'Fap@$-F\o6$-%$mapG6$%&denomG<#-FD6#FhnFK>Fhn-F ]t6%-%&numerG6#-FitFgx7#-Fat6$Fct/Fft;FfuFhsFit>Fhn7$-Few6$7#-Fat6$/FctFfuFbyFh n-Fat6$-%&coeffG6%FhnFctFNFby@$5-F\o6$FhnFJ4-FW6$Fhn-%%listG6#-%(polynomG6$F^qF K-F@6$%>invalid~recurrence~or~unknownGFCFhnF3F36(%(getnameGc(getname%:-65%)mini ndexGc)minindex%:-86%)maxindexGc)maxindex%:-85F3 M6R0 I&:-100R6'%#llG%"xG%%degsG%'nbiterG%%easyG6B%"yG%"iG%"LG%"nG%"jG%"lG%$indG%#l2G %&pivotG%%varsG%*leadcoeffG%"kG%*normalizeG%#lkG%.rationalcoeffG%#ttG%)finished G%%dminG%$degG%<ermG%#jjG%'lterm2G%%indsG%*polycoeffG%$ll2G%&dmin2G%%ind2G%#c oG%$ccoG%%cco1G%%cco2G%%cco3G6"FKC->8'-%%nopsG6#9$>8-7$-%$seqG6$&8$6#8%/Ffn;""" FN9%>8:-%&minusG6$-%'indetsGFQ<#Fjn@%/F\o<"C%>80R6#F&FK6$%)operatorG%&arrowGFKF RFKFKFK>82-%%typeG6$-F^o6$<#-%#opG6#-%$mapG6$FgpFR<#-%"OG6#Fin-%$setG6#%)ration alG>8;%&falseGC%>Fho%'normalG>F_pFfq>Feq-Fap6$Fcp-Faq6#-%(polynomG6$FcqF\o?(Ffn FinFinFN%%trueGC%>&8&Fen-%'seriesG6%*&FYFin)Fjn&9&FenFinFjn%)infinityG>&8)Fen&F RFen>&8?FenFin>84Ffq?(8(""!Fin9'4FisC4-%)userinfoG6&""$.T#%1iteration~numberGF[ t>85F`s?(FfnFinFinFNFdrC'>&8.Fen-Fho6#-%&coeffG6%FbsFjnF[t@%0F\uF\tC%>&8+FenFfn >86-Fgp6$-FP6#FgrFgr@'2FhtFju\2FjuFhtC&>FhtFju>87-Fgp6$,&F]vFin!""FinFgr>8=-%'l engthG6#Fbs>8>FfnC&>89Fgv@$2F]wF\wC$>FawFfn>F\wF]w?(88FinFinFN3-%$hasG6$Ffv&FZ6 #F[x-F^x6$FdwF`xFK@%531F[xFN4F]x332FNF[x0F\uF]q1-F^w6#&F]u6#8*-F^w6#F\uFav>FfvF dwC$>Fgu%%NULLG@%/F`uF\tFavC$>Fbs-Fjp6$FhoFbs@$30FbsF\t0-Fgp6$FinFbsF]qFav>FbyF fn>8,F`y@$/FgzF]q[@$FizC$>8/FbyFjz@$/FhtF`sFav>8<7#-FW6$FguFgn@&33F_p0FgzFin0Fg zFjvC$>83-%$absG6#-%)icontentG6#-%(convertG6$7#-FW6$*&F\uFinFYFin/FfnFb[l%"+G@$ 0F]\lFinC$>Fgz*&FgzFinF]\lFjv?&FfnFb[lFdr>F\u*&F\uFinF]\lFjvFeqC$>F]\l-%(conten tG6$Fd\lFT@$F^]lC$-%'divideG6%FgzF]\l.Fgz?&FfnFb[lFdr-F]^l6%F\uF]\l-%&evalnGFdy >F]\l&Fcs6#Faw?&FfnFb[lFdr@$0FfnFbyC$>Fbs-Fjr6%,&*&FgzFinFbsFinFin*&F\uFin&FcsF ayFinFjvFjnF`s@&5/FbsF\t/FczF]qC$>Fgr-Fjr6%,&*(FgzFinFfsFinFgrFinFin*(F\uFin&Fg sFayFin&FhrFayFinFjvFjnF`s>FisFdr/-Fgp6$""#FbsF[t>Fbs-%'subsopG6$/FinF\tFbs@$Fi sC%>FhtF`s?&FfnFb[lFdr@$3Fd_l2-F^wF^vFhtC$>FhtFaal>F^[lFfnFjz>Fb_l-Fjr6%*&FjnFi nF]\lFinFjnF`s>F]\l&FhrFg^l>8@&FgsFg^l?&FfnFb[lFdr@$Fj^l@'F_pC+>8B*&FgzFinFfsFi n>8C*&F^`lFin-%&numerGFdyFin>8A-%%igcdG6$FcblFfbl>Fcbl*&FcblFinF[clFjv>Ffbl*(F^ `lFinF\uFinF[clFjv>Fgr-Fjr6%,&*&FcblFinFgrFinFin*&FfblFinF_`lFinFjvFjnF`s>Fcbl- Fibl6#-Fb\l6#-%$addG6$*&-Fb\l6#-Fgp6$,&F[xFd`lFjvFinFgrFinF`xFin/F[x;Fin-%%iquo G6$F]vFd`l@$30FcblFin0FcblFjv>Fgr-Fjr6%*&FgrFinFcblFjvFjnF`s>Ffs*&F[clFinFcblFi nFeqC&>F[cl-%$gcdG6&Fdbl*&F\uFinF^`lFin.Fcbl.Ffbl>Fgr-Fjp6$FiqFdcl>Fcbl-Fh]l6%- Fe\l6$FgrFarFTF]fl@%F]elC$>Fgr-Fjr6%FfblFjnF`s>FfsFeel>FfsF[cl>Fgr-Fjp6$Fho-Fjr 6%,&*&FgzFinFgrFinFin*&F\uFinF_`lFinFjvFjnF`s>F_`lFfal>F^`lF\bl@$39(2*&%$BIGGFi nFNFin-%$minG6#-Fgp6#-Fjp6$F^w7#-FW6$FgrFgn-%'RETURNG6#%%FAILG-Fat6&""&Fdt%-cur rent~listG-%'lprintG6#Fdhl@$3/F[t,&F]tFinFinFinF^t-T%6%7#-FW6$Fb`lFgn%(numericG .F^[l>81-%(collectG6&-Fe\l6$&Fhr6#F^[lFarFT.%,distributedGFho@&F_p-F]^l6%F_jl-F b\l6#F_jl.F_jlFeq-F]^l6%F_jl-Fh]l6$F_jlFTF^[m-Fjp6$%'expandG7#-FW6$*&-Fau6$F_jl FYFin)Fjn,$F^sFjvFinFgnFKFK6&%&pade2Gc&pade2%:-33%,pade2/exminGc,pade2/exmin%:- 99FK M6R0 I%:-85R6%%$recG%"uG%"nG6"F(F(,&-%$maxG6#-%#opG6#-%$mapG6$F.-%'indetsG6$9$-%)spe cfuncG6$-%'linearG6#9&9%"""F=!""F(F(F(F( M6R0 I%:-52R6$%"tG%#YiG6"F'F'@%-%%typeG6$9$%'RootOfG-%)frontendG6%%%subsG7$/.%#_ZG9% -%#opG6#F,7$<$%"*G%"+G<#F3,&)F6-F86$""#-F86$FCF,""")-F86$FFF,-F86$FFFD!""F'F'F' F' M6R0 I3combstruct/agfeqnsR6'%$SysG%$AttG%$TypG'%(sizevarG%%nameG'%(VarlistG-%%listG6 #7$F*F*6>%$gfnG%$sysG%$typG%(varlistG%(attlistG%"iG%%varsG%#eqG%(nontermG%&rsid eG%(ep_listG%$lstG%$valG%$epsG%%uvalG%'AElistG%&cleanG%'redoneG%"AG%$feqG%"xG%$ attG%&checkG%%initG%)at_tableG%&indepG%-filtered_sysG%)ea_tableG6$%)rememberG%O Copyright~(c)~2000~INRIA.~All~rights~reserved.G6"C=@$0%7combstruct/InitializedG %%trueG-%0combstruct/initGFQ-%'forgetG6$%2combstruct/NameOfG/%-subfunctionsG%&f alseG>8$-%8combstruct/checkgrammarG6$9$9&>8%&F[o6#""">8&&F[o6#""#>8'9(-%)userin foG6&FjoF"Q.grammar~used:FQFbo>8.-%'selectG6$R6#FFFQ6$%)operatorG%&arrowGFQ-%%t ypeG6$-%#opG6$FjoF_oF.FQFQFQ9%?&8/FcpFVC&>80-F`q6#-F`q6$FjoFdq>81-F`q6$FeoFdq@$ 0-F`q6$""!F]r(%6combstruct/attribute/GFgqY6$QAInvalid~attribute~production,~%1F QFdq?(8)FeoFeo-%%nopsG6#F]rFV@%4-%)assignedG6#&826#-F`q6$F[sF]r>FdsFgq>Fds*&Fgq FeoFdsFeo>Fcp<#-F`q6#-%$mapG6$RFhpFQFipFQ-F`q6#-F`q6$FeoF_oFQFQFQFcp>8;-%'remov eG6$RFhpFQFipFQF\qFQFQFQFbq>Fit-%9combstruct/checkagrammarG6$FboFit>89&FitFdo>8 (&FitFio>8<&Fit6#""$-F_p6&FjoF"Q8attribute~grammar~used:FQFcu@$3/9#""'-F]q6$&9" 6#FdvF*-%'assignG6$&Fhv6#Fcv7$FboFcu>8:-%Ffu&FawFdo>F\p&FawFio>8*&FawF[v>%;combstruct/gfeqns/indcountGFeo>8=<">8 >-%-selectremoveG6$RFhpFQFipFQ5/-%$rhsG6#F_o.%%AtomG/Fjx.%(EpsilonGFQFQFQFbo?&8 +&FcxFdoFVC'>8,-F`q6$FeoFcy>8--F`q6$FjoFcy>&876#Fgy-%(combineG6%-%;combstruct/a gfeqns/findeqnG6(F[z&FiuFazFgoF\pF[xFfu%&powerG%)symbolicG@$-%'memberG6$FgyFcp> F_z*&F_zFeo&FesFazFeo>&8?Faz-%(unapplyG6$F_z-F`q6#F\p?&Fcy&FcxFioFVC)>FgyFhy>F[ zF\z>&Fhz6#F`y&FiuFa\l>&Fhz6#F]y&FiuFe\l>F_z-Fcz6%-Ffz6)F[zFhzFgoF\pF[xFfuFd[lF izFjz>F`\l%%NULLG>Fd\lF]]l7#-%$seqG6$/-866#Fh[l&F`z6#Fe]l/Fe]l-%%map2G6%F`qFeoF boFQ6%F^x%.combstruct/nbGFUFQFQ M6R0 I8%7$7 $9&.%%sizeG-%#opG6#9'>8'7#-F=6#-%$mapG6$F=-%%map2G6%%'subsopG/"""%%NULLGF6>8+-% (convertG6$FA%$setG>8(-%&minusG6$-FR6$9(FTFP@$0FV<"Y6$QXthe~following~attribute s~must~be~assigned~variables,~%1F3FV>8$FA@$0-%*intersectG6$FP-%'selectG6%%%type G9$%%nameGFinYQRvariables~must~only~be~associated~with~attributesF3?&8*F6%%true G>&8&6#-F=6$""#F]p-F=6$FMF]p>F67#-F=6#-FF6$R6#%"xGF36$%)operatorG%&arrowGF3-F=6 $FMFhoF3F3F3F6@$0-%%nopsG6#F6-Fiq6#<#-F=FjqYQDvariables~must~be~uniquely~assign edF3>8)-Fbo6$-%'indetsG6$FhoFio-FR6$F6FT@$0FbrFinY6$Qgnnon-terminal~names~canno t~also~be~used~as~variable~names,~%1F3Fbr7%F^oF6FapF3F3F3F3 M6R0 I;combstruct/agfeqns/findeqnR6)%&rsideG%)at_tableG%$typG%(varlistG%%varsG%'atli stG%)ea_tableG67%(sizevarG%"FG%$attG%,newat_tableG%&constG%%termG%%FposG%#wtG%' argseqG%"rG%$PhiG%)num_argsG%$phiG%)multfactG%#AzG%$indG%"uG%"kG%"iG%"jG%%findG 6#%]oCopyright~(c)~2000~by~Marni~Mishna~and~INRIA.~All~rights~reserved.G6"C%>8$ &9'6#"""@&-%%typeG6$9$%%nameGC$@'/FP.%(EpsilonG>8--%$mulG6$)&9(6#8%-%7combstruc t/agfeqns/powG6$&9%Fin-F\o6$&&F_o6#FUFin""!/Fjn9)/FP.%%AtomG>FX-FZ6$)Fgn-F\o6$F ^o-F\o6$&&F_o6#FioFinFKFfoC'?&FjnFgo%%trueGC%>8&F^o>&8(FinFeo@%-FN6$F[q.%(index edG>&8'6$FjnFjnFKC%>F[q-%'expandG6#F[q@$4-FN6$F[q%"+G>F[q7#F[q?&8)F[qFhp@'-%)pa tmatchG6%Fer*&'.%$wt0G%'atomicGFK'.%#G0G%)functionGFK.88C%>8*-%#opG6$Feo-%%subs G6$FdsF`s>8+-F\t6$FdsF\s>&Ffq6$FjnFgsF_t-FN6$Fer.%*algebraicGC$@$-%'memberG6$-F jn6#FP-%'indetsG6#FerY6$Q8term~is~non-linear:~%1~FDFer>F]q,&F]qFKFerFKY6$Q=inco rrect~grammar~format,~%1FDFer>8,-%$seqG6$-FZ6$)Fgn-F\o6$FctFeoFfo/FgsFgo@%/F\v- F^v6$FK/86;FK-%%nopsG6#Fgo>FXFK>FX-FP6#F\v@$5-FN6$FP%(epsilonG-FN6$FP%%atomG>FX -&9*F_u6#-Fis6#FX>FX*&-FZ6$)Fgn-F\o6$F]qFeoFfoFKFXFKOFX-FN6$FPFbsC)>8.-Fis6$Feo FP>8/-F^wF_u?&FjnFgoFhpC%>F[qF^o>F]qFeo@)F`q>&FfqFin7#-F^v6$F[q/F[w;FKFcy-Fhr6% F[q,&'.%%phi0GFbsFK'.%'const0GFgtFKFcsC&>80-F\t6$FdsFez@$0-Fis6$FeoF\[lF_yY6$Fj uF[q>F]q,&F]qFK-F\t6$FdsFhzFK>F[z7#-Fis6#F\[l-FN6$F[qFgtC$>F]q,&F]qFKF[qFK>F[z7 #-%"$G6$FeoFcyYFd[l>&FfqFdo7#-Fe\l6$FcoFcy>&FfqFep7#-Fe\l6$FdpFcy>81-FZ6$)FgnF] qFfo@'/F_y.%&UnionGO*&Fc]lFK-%$addG6$-9!6%-Fis6$F[wFP-%%map2G6%FisF[wFfq&9"6#;" "$9#/F[w;FKFdyFK/F_y.%%ProdGO*&Fc]lFK-FZF_^lFKC%>82-Fa^l6%-Fis6$FKFP-Ff^l6%FisF KFfqFh^l@$/Fh_lFKOFc]l@3/F_y.%&CycleG>FX-%9combstruct/agfeqns/CycleG6&FP9&Fh_lF I5/F_y.%$SetG3/F_y.%)PowerSetG/Fj`l.%)labelledG>FX-%7combstruct/agfeqns/SetG6'F PFj`lFh_lFIFKF`al>FX-Fhal6'FPFj`lFh_lFI!""/F_y.%)SequenceG>FX-%FX-F\t6$/FGFh_l-Fa ^l6$-Fis6$""#FP&Fi^l6#;FcclF]_l-FN6$F_y.%&deltaG@%FcalYQOdelta~should~not~be~in ~a~labeled~specificationFD@%-FN6$F_y&%1combstruct/DeltaG6#%(integerG>FX-F\t6$/F G)%$varG-Fis6#F_yFh_lC&>83&876#%;combstruct/gfeqns/indcountG>Fcel,&FcelFKFKFK@+ /F_y&Fbdl6#F^alC$>84FK>85FK/F_y&Fbdl6#Fe`lC$>F\fl-&%*numtheoryGF\\l6#F_el>F^flF K-FN6$F_y&Fbdl6#-F^alFcdlC$>F\flFK>F^fl-Fis6#F[el-FN6$F_y&Fbdl6#-Fe`lFcdlC$>F\f lFdfl>F^flFaglYQDunknown~Delta~type~in~specificationFD>FX-%$SumG6$*&F\flFK-F\t6 $-%FX*&FGFK-%%diffG6$Fh_lFGFK/F_y%/combstruct/IntG>FX-%$intG6$*&Fh_lFK FGF]blFGY6$Q8$-%#opG6$7$""#""!F3>8%-F:6$7$F=F=F3@'/F@.%%cardGC $>F@-F:6$7$F=F4F3,&F5F4-%$addG6$*&)9%8&F4-9'6#FTF4/FT;F>,&F@F4!""F4Fen/F8%"=G*& )FSF@F4-FV6#F@F4-FO6$FQ/FT;F>F@F-F-F-F- M6R0 I9combstruct/agfeqns/CycleR6&%&rsideG%$typG%#AzG%(varlistG6*%"iG%"kG%#gfG%$allG %"uG%"jG%$indG%(cardtypG6#%^oCopyright~(c)~1995~by~Eithne~Murray~and~INRIA.~All ~rights~reserved.G6"@%/9%.%)labelledG-%!""FER6#%"xGF46$%)operatorG%&arrowGF4-%#ifG6%/F=""!FP*$F=FEF4F4F 4C%>8*&8)6#%;combstruct/gfeqns/indcountG>8&*(-&%*numtheoryG6#%$phiG6#FTFDFTFE-F @6#*$,&FDFD*&)8(FTFD-%%subsG6$-%FDFEFEFD@% /-%%nopsG6#F=FDC$>FX,&FXFDFDFD-%$SumG6$-Fdo6$/FboFDFZ/FT;FD%)infinityGC%>8+-%#o pG6$7$""#FPF=>8%-F_q6$7$FbqFbqF=@'/Fdq.%%cardGC&>FXFap>Fdq-F_q6$7$FbqFDF=>8'Fbp ,&FcrFD-Fdo6$Fgp-%(convertG6$-%'seriesG6%,&-%$addG6$-%%evalG6#-Fdo6$/FT8$FZ/Fgs ;FD,&FdqFDFEFDFD-%"OG6#)FboFdqFDFbo,&%&OrderGFD""$FD%(polynomGFE/F]q%"=G-%&coef fG6%-F[s6%,&-F_s6$Fas/Fgs;FDFdqFD-F\t6#)Fbo,&FdqFDFDFDFDFboF_tFboFdq-Fdo6$Fgp-F hr6$FhtFbtF4FWF4F4 M6R0 I7combstruct/agfeqns/SetR6'%&rsideG%$typG%#AzG%(varlistG%&constG6*%"iG%"kG%"uG% $allG%+firsttermsG%"jG%$indG%(cardtypG6#%OCopyright~(c)~2000~INRIA.~All~rights~ reserved.G6"@%/9%.%)labelledG-%""!"""*$-%*factorialG6#F>!""F5F5F5@%/- %%nopsGFRFNC%>8*&8)6#%;combstruct/gfeqns/indcountG>Fhn,&FhnFNFNFN-FA6#-%$SumG6$ *&-%%subsG6$-%8+-%#opG6$7$""#FMF>>8%-Fcp6$7$FfpFfpF>@'/Fhp.%%cardGC(>Fhp-Fcp 6$7$FfpFNF>>FZFen>FhnFjn>8'F[o@'/FhpFM>8(FM/FhpFN>F\rFN>F\r-Fbo6$/8&FN-%(conver tG6$-%'seriesG6%-FA6#,&-%$addG6$*()Fcr8$FNFbsFS-Fbo6$-Feo6$FgoFbs*&)Fjo,&FbsFNF NFNFNF?FNFN/Fbs;FN,&FhpFNFSFNFN-%"OG6#)FcrFhpFNFcrF^p%(polynomG,&FhqFNF\rFS/Fap %"=G@%FjqFN-%&coeffG6%-Fhr6%-FA6#,&-F^s6$F`s/Fbs;FNFhpFN-F^t6#)Fcr,&FhpFNFNFNFN FcrF^pFcrFhp@%FjqFN-Fbo6$Fbr-Fer6$FitFatF5FgnF5F5 M6R0 I7combstruct/agfeqns/powR6$%"xG%"yG6"F'F'-%#ifG6%4-%%typeG6$9$%(indexedGF/9%F'F 'F'F' M6R0 I8$<#-%#opG6#9$>&8%6#""! FA?(8)FK""",&9%FN!""FN%%trueGC$>&FI6#FM-9!6$FAFM@$/FU%%FAILGOFfn>8*-%%map2G6%FD 7$FNFKFF>8+7#-FD6#-FD6$7$FNFNFF@$0-%%nopsG6#<#-FD6#-F[o6%FDFeoFFFNYQVInconsiste nt~format.~The~first~variable~must~be~size.F>?(8,FNFNFPFRC'>8--%$mapG6%%"$GF_oF cp>8(-&%+combstructG6#%+allstructsG6$-%,CombinationG6#Ffp/.%%sizeGFcp>8.-%%subs G6$<#-%$seqG6$/&F_oFVFM/FM;FN-Fio6#F_oF\q>&8'6#Fcp-%&unionG6$<#-F`r6$-F`r6$/-&% "DG6#-FD6#846#82&F]tFhs/F]tFin/F[tFjq<#-F`r6$-F`r6$/-%%diffG6$-F]t6#-FDFgr-FD6# &F\q6#85-&F]t6#-FD6#&FjqF`uF[uF_t/Fau;FN-Fio6#F\q>&83F[s<#-F`r6$-F`r6$/-F^t6$&F _o6#FN-Fjp6$FN,&FfrFNFQFN-F^t6#FgvF`tF_t>8/<#-F`r6$/80FN/Fdw&F_o6#;""#Ffr>&F^vF J<#-F`r6$/-F]tFfv-F]tF]wF_t@%/FPFK>81-F\r6%F_wF[xFHC$>8&<#-F`r6$-FD6#Fir/Fcp;FN FP>Fex-F\r6'FjxF_w<#-F`r6$-FD6#&F^vFV/FM;FKFP<#-F`r6$-FD6#FU/FM;FKFO<#-F`r6$-FD 6#-Fht6$FF-FD6#Fdw/FdwF\q-%)userinfoG6%Fiw%/agfmomentsolveGFex@$3/9#""$-%%typeG 6$&9"6#Ff[l%%nameG-%'assignG6$Fj[l7#-FD6#Fex%3combstruct/gfsolveG-%F>F>F> M6R0 I5combstruct/agfseriesR6'%$sysG%$attG%$typG'%$varG%%nameG%(VarlistG6$%$eqsG%(va rlistG6#%enCopyright~(c)~Marni~Mishna~and~INRIA.~All~rights~reserved.G6"C&Z%>8$ -%3combstruct/agfeqnsG6#9"F1YF1>8%7#-%#opG6#-F?6$"""&F56#FC%4combstruct/gfserie sG-%;combstruct/gfseries/seriesG6&F59&9'F8(-%6combst ruct/parse_argsG6#9"@%-%%typeG6$&F86#"""-%*identicalG6#.%(grammarG-%>combstruct /allstructs_grammarG6$&F86#""#&F86#""$C(>8)FL>8*FO@'3-F?6$FT.%)functionG-F?6$-% #opG6$""!FT.%%nameGC$>8%Fin>8&6$FV-Fjn6#FT-F?6$FTF]oC$>FaoFT>FcoFVY6$Q:invalid~ structure~name~%16"FT>8'(%7combstruct/allstructs/GFao@$4-F?6$-%%evalG6$FapFC.%* procedureGY6$QAnot~implemented~for~structure~%1F_pFT-Fap6#FcoF$F$F$F_p M6R0 I>combstruct/allstructs_grammarR6$%%specG%$sizG6(%(nontermG%$sysG%$typG%&totalG %$allG%"xG6#%fnCopyright~(c)~1996~Waterloo~Maple~Inc.~All~rights~reserved.G6"C) >8$&9$6#""">8%&F56#""#>8&&F56#""$>8'-&%+combstructG6#%&countG6$7%F3F9F>/.%%size G9%>8(<"?(F0F7F7F02-%%nopsG6#FPFC>FP-%&unionG6$FP<#-%;combstruct/draw/tagged_ob jG6$FJFN7#-%$seqG6$-%8$-%&minusG6$-%%map2G6%%#opG""!-%'indetsG6$9%%)functio nG-F@6%FBFC-FE6$9$FH>8%-F@6%FB7$"""FCFG@$-%'memberG6$.%%sizeGFOYQTattribute~nam e~'size'~reserved~for~number~of~atoms.F8>8&-FE6$F;<$.%*protectedG%)constantG@$0 Fgn<"Y6$QAthe~names,~%1,~are~used~by~MapleF8Fgn>8)-F@6%FBFSFM?&8/FG%%trueGC%>8' -FB6$FRFio>8(-FB6$7$FSFSFio@'4-%%typeG6$&&8*6#Fap6#F]p%(indexedGY6%QNattribute~ function~for~%1~%2~multiply~definedF8F]pFap45-FV6$F]pF;-FV6$FapFeoY6%Q`oattribu te~function~for~%1~%2~does~not~correspond~to~a~valid~structureF8F]pFap>Fjp-FB6$ ""#Fio>8+FM?&83FarFjoC(>8.-FB6$F_rFcr>Fap-FB6$FSFcr?&F]pF;FjoC$>8,Fjp@%4-Fhp6$F _s.F_q>&8-F^q/-F]pF]q-%@combstruct/checkagrammar/verifyG6%FfrF_s.%'newattG>Ffs% %NULLG>80<#-%$seqG6$Ffs/F]pF;Z%>82-%Acombstruct/checkagrammar/flattenG6&FcrFbtF ;F[qF8YFjt>&81F]qFjt7%<#-Fet6$-FB6#F`u/FapFeoF;F\qF8F8F8F8 M6R0 I@combstruct/checkagrammar/verifyR6$%&rsideG%&atproG6(%"rG%$PhiG%'numposG%)phi_ exprG%&constG%"iG6"F.C%>8(-%'expandG6#9%@'-%%typeG6$9$.%%nameG@%5/F:.%(EpsilonG /F:.%%AtomGC$@$4-F86$F5.%*algebraicGY6$QUthe~value~of~Atoms~and~Epsilons~must~b e~constant,~%1F.F5>8$F5>FPF5-F86$F:%)functionGC'>8%-%#opG6$""!F:@%-F86$FW.%(use cardG>8&""">F\o-%%nopsG6#F:>8'-%Acombstruct/checkagrammar/collectG6%FWF1.F1@%/F coFen>FPFenC$@$0-F`o6#FcoF\oY6%QDincorrect~specification,~%1,~for~%2F.F5F:>FP-F W6#-%$seqG6$-9!6$-FY6$8)F:-FY6$F^qFco/F^q;F]oF\o>FP,&FPF]oF1F]oY6$QAunknown~gra mmar~construction,~%1F.F:FPF.F.F.F. M6R0 IAcombstruct/checkagrammar/flattenR6&%&gprodG%&atsetG%(attlistG%)at_tableG65%"F G%"AG%&rsideG%$PhiG%%FrhsG%'fuldepG%$depG%#f1G%%restG%(empty_FG%&emptyG%$ordG%$ oldG%#okG%%old2G%'atsubsG%&atproG%+card_restsG%&constG6"F=C1>8%-%#opG6$"""9$>8& -FB6$""#FE>8'-FB6$""!FG?&8$9&%%trueGC$>8(&9'6#FQ@%-%%typeG6$FV%(indexedGC'>&8)F YFO>&8*FY<">8+R6$%%exprGF*6%%"QG%)new_exprG%#crGF=F=C%>T#-%'indetsG6$FE<$%"=G%" F@-%%subsG6$-%$seqG6$/FG%%NULLG/FGF\pFE-Fep6$<#-Fhp6$/FQ-9%FY/FQ-F^p6$FE.%%n ameGF@F=F=6$F;85>&8,FY-Fep6$<$/-FQ6#.%%AtomGFD/-FQ6#.%(EpsilonGFO-Fbo6$FHFQ>&8- FYFQC&>F]r-%Acombstruct/checkagrammar/collectG6%FLFV.86>F[oFgs>F^o-%%map2G6%FBF O-F^p6$F[o-%)functionG6#-%*identicalG6#F@>F_sF[q>8.<#-Fhp6$F_s/FQFR>8/7#-FB6#Fg t>80FR>81Fgt?(F=FDFDF=082FduC%>FguFbu>FbuFdu?&FQ-%&minusG6$FRFduFS@%/F^oF`oC$>F du-%&unionG6$Fdu<#FQ>F]u7$-FB6#F]uFQ>F^o-F]v6$F^oFdu@$0FduFRY6$Qfnthe~following ~attributes~have~unresolvable~dependencies,~%1F=-FB6#F\v>83F`o?&FQF]uFSC$>F[o,& F]rFD-Fep6$FfwF[oFD>Ffw-Fdv6$Ffw<#/-FQFdtF[o>Ffw-F]v6$Ffw<#-Fhp6$Fax/FQFgt?&84F fwFS>&FX6#-FB6$7$FDFOF[y-%@combstruct/checkagrammar/verifyG6$FG-FB6$FJF[y<#-Fhp 6$/FbxFW/FQ-F[t6%FBFayFfwF=F=F=F= M6R0 IAcombstruct/checkagrammar/collectR6$%%termG%%exprG6+%"fG%&otherG%%mainG%#exG%" aG%&foundG%$nexG%"iG%"jG6"F1C%@%-%%typeG6$9%.%'atomicGC$>8&""!>8%F7C(@%-F56$F7< $%"*G.%)functionG>8*7#F7>FIF7>8+F=?&8'FI%%trueGC$>FM,&FM"""FTFT@%-%)patmatchG6% FO*&'.%#a1GF9FT'.%#f1GFGFT.8)C%>8$-%%subsG6$F[oFhn>8(-F`o6$F[oFen@%/-%#opG6$F=F ^o9$C$>&F<6#FM-%$mapG6%FE7#-Fio6#F^oFco>&F?F_pF=C$>F^pF=>FgpFOC$>F^pF=>FgpFO>F< -%$addG6$&F<6#8,/Fdq;FTFM>F?-F`q6$&F?FcqFeq@$0FF<-F[p6#-Fio6#F<@$/9#""$-%'a ssignG6$&9"6#FerF?F%1combstr uct/atomsG-%&unionG6$<#.%%AtomG-%%map2G6%%#opG"""-%'indetsG6$9$/F+-%*identicalG 6#F<>%4combstruct/epsilonsG-F96$<#.%(EpsilonG-F?6%FAFB-FD6$FF/F+-FI6#FPF2YQLInc orrect~grammar~format~for~Atoms/EpsilonsF2>8$-%*traperrorG6#-%9combstruct/check grammar1G6$FF9%@$/Ffn%*lasterrorGY&%.lastexceptionG6#;""#!"">F7-F96$F;-F?6%FAFB -FD6$&Ffn6#FBFG>FL-F96$FO-F?6%FAFB-FD6$F_pFVFfnF26$F7FLF26" M6R0 I9combstruct/checkgrammar1R6$'%$SysG-%$setG6#/%%nameG%)anythingG%$TypG6,%$typG% $sysG%$valG%%indsG%)conflictG%"iG%,badnontermsG%"xG%$occG%'doubleG6%%)rememberG %'systemG%fnCopyright~(c)~Eithne~Murray~and~INRIA.~All~rights~reserved.G6"C6@$/ 9$<"YQ8$.%)labelledG/FH.%*unlabe ledG>FL.%+unlabelledG>FLFH@$30FLFM0FLFSYQ_psecond~argument~must~be~either~of~'l abeled',~'unlabeled',~'labelled'~or~'unlabelled'F=>8'-%'indetsG6$FAF+>8(-%*inte rsectG6$<$-%$seqG6$(%&gArg_G8)/Ffo;"""""$-Fbo6$(%&gLoc_GFfo/Ffo;Fio""*Fgn@$0F\o FBY6$QTthe~names~%1~are~reserved~for~the~combstruct~systemF=F\o>F\o-Fin6$Fgn<$. %)constantG.%*protectedG@$FcpY6$Q?the~names~%1~are~used~by~MapleF=F\o>8,<#-Fbo6 $-%'selectG6%%%typeG-%$mapG6$%$lhsG7#-%#opG6#FA-%*identicalG6#8+/FgrFgn>8--%&mi nusG6$-F\s6$Fdq-F]r6$R6#F6F=6$%)operatorG%&arrowGF=7#FAF=F=F=Fgn<#7"@$0FjrFBY6$ QJthe~names,~%1,~are~defined~more~than~onceF=<#-Fbo6$-Fbr6$FioFgr/FgrFjr>8%-%Bc ombstruct/standardform/usecardNFGFcr@$30-Fin6$FA-Fer6#.%"ZGFB4-%(hastypeG6$FA/F ^uF,>Fft-%&unionG6$Fft<#/F`u.%%AtomG-%)userinfoG6&Fjo%+combstructG%>first~norma l~form~of~grammar:GFft>8&-%Gcombstruct/wellfounded/find_valuationsG6#Fft>8*<#-F bo6$-F_rFfr/Fgr-Fin6$-Fbr6#-%%evalG6#Fev/F+-Fer6#%)infinityG@$0FjvFBY6$QH%1~pro duce(s)~no~objects~of~finite~sizeF=Fjv@$4-%;combstruct/is_well_foundedG6$FftFev YQFj v-%@combstruct/wellfounded/PowerSetG6$FftFL@$F\xYF^x7$FftFLF=F=F=FE M6R0 IBcombstruct/standardform/usecardNFR6#%$SysG6#%$sysG6#%fnCopyright~(c)~Eithne~M urray~and~INRIA.~All~rights~reserved.G6"C$>8$-%%evalG6#-%%subsG6$<&/%&CycleG%@c ombstruct/standardform/CycleNFG/%$SetG%>combstruct/standardform/SetNFG/%)PowerS etG%Ccombstruct/standardform/PowerSetNFG/%)SequenceG%Ccombstruct/standardform/S equenceNFG9$-F26$7&/F:.F9/F@.F?/F7.F6/F=.Fcombstruct/standardform/SetNFR6$%"AG%%exprG6#%(newcardG6#%fnCopyright~(c)~Eit hne~Murray~and~INRIA.~All~rights~reserved.G6"@'/9#""!YQ6Set~takes~an~argument6" /F."""-%$SetG6#9$C$>8$-%Bcombstruct/standardform/checkcardG6#9%-F66$F8F;F+F+F+F 2 M6R0 ICcombstruct/standardform/SequenceNFR6$%"AG%%exprG6#%(newcardG6#%fnCopyright~(c )~Eithne~Murray~and~INRIA.~All~rights~reserved.G6"@'/9#""!YQ;Sequence~takes~an~ argument6"/F."""-%)SequenceG6#9$C$>8$-%Bcombstruct/standardform/checkcardG6#9%- F66$F8F;F+F+F+F2 M6R0 I@combstruct/standardform/CycleNFR6$%"AG%%exprG6#%(newcardG6#%fnCopyright~(c)~E ithne~Murray~and~INRIA.~All~rights~reserved.G6"@'/9#""!YQ8Cycle~takes~an~argume nt6"/F."""-%&CycleG6#9$C%>8$-%Bcombstruct/standardform/checkcardG6#9%@(-%&evalb G6#/7#F;7#/.%%cardGF/Y6$QBthere~are~no~cycles~of~size~0:~%1F+-F66$F8F?-FB6#/FE7 #1F/FH>F;1F4FH-FB6#/FE7#1FHF4>F;/FHF4-F66$F8F;F+F+F+F2 M6R0 ICcombstruct/standardform/PowerSetNFR6$%"AG%%exprG6#%(newcardG6#%fnCopyright~(c )~Eithne~Murray~and~INRIA.~All~rights~reserved.G6"@'/9#""!YQ;PowerSet~takes~an~ argument6"/F."""-%)PowerSetG6#9$C$>8$-%Bcombstruct/standardform/checkcardG6#9%- F66$F8F;F+F+F+F2 M6R0 IBcombstruct/standardform/checkcardR6#%%exprG6$%"kG%"rG6#%fnCopyright~(c)~1996~ Waterloo~Maple~Inc.~All~rights~reserved.G6"C%@+2"""9#Y6$Q@bad~cardinality~restr iction:~%1F+9"-%*typematchG6$9$2'8$%*nonnegintG-%*identicalG6#.%%cardG>8%1,&F;F /F/F/F@-F66$F82F='F;%'posintG>FC1F@,&F;F/!""F/-%%typeG6$F8<%1FFCF 8Y6$F3F8@$/7#FC7#1F@""!>FC/F@FhnFCF+F+F+6" M6R0 I@combstruct/wellfounded/PowerSetR6$%$SysG%$typG6,%#eqG%(nontermG%&rsideG%"xG%% eqnsG%'eqnvalG%%sersG%$sysG%%varsG%)ea_tableG6#%fnCopyright~(c)~1996~Waterloo~M aple~Inc.~All~rights~reserved.G6"C,>8+-%%evalG6#-%%subsG6$/%)PowerSetG%Bcombstr uct/wellfounded/changecardG9$>%;combstruct/gfeqns/indcountG"""?&8$-%$mapG6$%$lh sG-%'selectG6$R6#F+F46$%)operatorG%&arrowGF4/-%#opG6$""#FA.%(EpsilonGF4F4F4F7%% trueG>&8-6#FFFD?&FF-FH6$FJ-FL6$RFOF4FPF4/FT.%%AtomGF4F4F4F7FZ>FfnR6#%"zGF4FPF4F AF4F4F4>&Fgn6#F`oRFdoF4FPF4FAF4F4F4>&Fgn6#FXFD?&FFF7FZC&>8%-FU6$FDFF>8&-FU6$FWF F>&8,6#.%%sizeG.Feo@$50FdpFX4-%)assignedG6#&8)6#F`p>Feq-%;combstruct/agfeqns/fi ndeqnG6)Fdp.%)at_tableG9%7#F]qFip<#F[qFgn>8(7#-%$seqG6$/-8'6#F]q&Ffq6#Fir/Fir-F H6$FU7#-%(indicesG6#Ffq>8*-%;combstruct/gfseries/seriesG6&FbrF^rF]qF]q<#-Fer6$- %#ifG6%-%&evalbG6#/&FesF\s""!-FU6$FdtFir%%NULLG/Fir-FH6$FU7#-Fbs6#FesF46#FCF46" M6R0 IBcombstruct/wellfounded/changecardR6$%"AG%$resG6#%"kG6#%fnCopyright~(c)~1996~W aterloo~Maple~Inc.~All~rights~reserved.G6"@%3/9#""#-%*typematchG6$9%1'8$%(integ erG-%*identicalG6#.%%cardG-%)PowerSetG6$9$/F8,9$>&8$6#.%(EpsilonG""!>&F:6#.%%AtomG"""? &8'F6%%trueG@'/-%#opG6$""#FFF&F:6#-FK6$FDFFF>>F6-%&minusG6$F6<#FF/FJFBC$>FP FD>F6FU>FP%)infinityG>8+FG?(F3FDFDF3FjnC$>Fjn%&falseG?&FFF6FGC(>8&FR>8%FJ>8(&F: 6#Fbo>8)-%)simplifyG6$-%@combstruct/wellfounded/find_valG6$FdoF:Fhn>8*-%$minG6$ FfoFjo@$0FbpFfoC%>FgoFbp-%)userinfoG6(""$%+combstructG%7new~valuation~for~typeG Fbo.%#isGFgo>FjnFG-%%evalG6#F:F3F3F36" M6R0 I@combstruct/wellfounded/find_valR6$%&rsideG%$valG6(%"rG%$PhiG%"kG%#r1G%#r2G%#r sG6#%fnCopyright~(c)~1996~Waterloo~Maple~Inc.~All~rights~reserved.G6"C$@+/9$.%( EpsilonG>8$""!/F4.%%AtomG>F8"""-%%typeG6$F4%%nameGC$@$4-%)assignedG6#&9%6#F4Y6$ Q;undefined~non-terminal:~%1F0F4>F8FI-F@6$F4%)functionGC$>8%-%#opG6$F9F4@1/FU.% &UnionGC$?&8)7#-FWFK%%trueG>&FJ6#Fin-9!6$FinFJ>F8-%$minG6#-%$seqG6$F^o/FinFjn-F @6$FU.%(productGC%@$2-%%nopsGFK""#Y6$QDProd~takes~at~least~2~arguments:~%1F0F4? &FinFjnF\o>F^oF`o>F8-%(convertG6$7#Fgo%"+G-F@6$FU.%(usecardGC%>F8-Fao6$-FW6$F>F 4FJ>&FJ6#FhqF8@%/F8F9@%30FU.%)PowerSetG5/FbpF>-F@6$-FW6$FdpF4-%%evalG6#1%(integ erG-%*identicalG6#.%%cardGY6$QM%1~derives~infinitely~many~objects~of~size~0F0F4 Y6%Qip%1~produces~an~element~of~size~0~inside~%2~which~is~not~allowed.~See~comb struct[specification]F0F4FU@&5Fer-F@6$Fhr-F[s6#1F_sF^s@$-%'memberG6$FU7%.%$SetG Fbr.%)SequenceG>F8F9-%*typematchG6$Fhr<$/F_s'8&F^s1F`uF_s@%/FauF9>F8F9>F8*&F8F> FauF>/FU.%&SubstGC%@$0FbpFdpY6$Q>Subst~takes~two~arguments:~%1F0F4>8'Ffq@%/FbvF 9Y6$QHcannot~substitute~objects~of~size~0:~%1F0F4C'>8(-Fao6$FhrFJ@$/FjvF9Y6$QMc annot~substitute~into~objects~of~size~0:~%1F0F4>F[rFbv>&FJ6#FhrFjv>F8*&FjvF>Fbv F>-F@6$FU.%&deltaGC'@$0FbpF>Y6$Q7too~many~arguments:~%1F0F4>F8-Fao6$F[oFJ>&FJ6# F[oF8@$4-F\u6$FU&%1combstruct/DeltaG6#<$F`u-%)specfuncG6$F`u<$Fft.%&CycleG>FauF >>F8Fgu-Fct6$FU7$%1combstruct/ThetaG%/combstruct/IntGC&@$F^xYF`x>F8Fcx@$F^rY6%Q in%1~must~not~have~arguments~that~produce~elements~of~size~0:~%2F0FUF4>FfxF8/FU .%0combstruct/PSetGC%>F[rFfq>FdwF[w>F8F[rY6$Q8unknown~constructor:~%1F0FUY6$Q7i nvalid~production:~%1F0F4F8F0F0F06" M6R0 IBcombstruct/wellfounded/properties=6"%&falseGE\[l'%$SetG7%""!%#2aG7#%"PG%)Powe rSetGF(%/combstruct/IntG7%R6#%"nGF$6$%)operatorG%&arrowGF$-%$maxG6$"""9$F$F$F$% #2bG7$%"QGF,%)SequenceGF(%1combstruct/ThetaG7%RF1F$F3F$F6F$F$F$F;F<%&CycleG7%RF 1F$F3F$F:F$F$F$F*F+6" M6R0 IIcombstruct/wellfounded/MultiConstructors7(%$SetG%)PowerSetG%/combstruct/IntG% )SequenceG%1combstruct/ThetaG%&CycleG6" M6R0 I=combstruct/wellfounded/N_Phi=6"%&falseGE\[l'%$SetG""!%)PowerSetGF(%/combstruc t/IntG"""%)SequenceGF(%1combstruct/ThetaGF+%&CycleGF(6" M6R0 I@combstruct/wellfounded/propertyR6$%$PhiG%"nG6#%"lG6#%_pCopyright~(c)~1995~by~ Eithne~Murray,~Paul~Zimmermann~and~INRIA.~All~rights~reserved.G6"C$>8$-%#opG6$" "$&%Bcombstruct/wellfounded/propertiesG6#9$-F06$-%$minG6$,&9%"""F>F>-%%nopsG6#F .F.F+F+F+6" M6R0 I;combstruct/is_well_foundedR6$%$sysG%$valG6)%"tG%$PhiG%"NG%"nG%"gG%"lG%#eqG6#% _pCopyright~(c)~1995~by~Eithne~Murray,~Paul~Zimmermann~and~INRIA.~All~rights~re served.G6"C.>8&""!?&8*9$%%trueGC$>8$-%#opG6$""#F7@$-%%typeG6$F<%)functionGC$>8% -F>6$F5F<@(-%'memberG6$FH%Icombstruct/wellfounded/MultiConstructorsG>F4-%$maxG6 $F4&%=combstruct/wellfounded/N_PhiG6#FH-FC6$FH.%&deltaG>F4-FR6$F4"""/FH.%&Subst G>F4-FR6$F4&9%6#-F>6$FhnF<-%)userinfoG6&F@%+combstructG%OThe~dependency~graph~i s~invariant~up~from~sizeGF4@$2F5F4-Feo6&F@Fgo%=Looking~for~loops~for~size~8(-%Hcombstruct/wellfounded/dependency_graphG6%F8F_pF`o-Feo6 (""$Fgo%>the~dependency~graph~for~sizeGF_p.%#isGFcp?(F1FhnFhnF1F9C$>8)-%Acombst ruct/wellfounded/has_cycleG6$FcpF8@%-FC6$F`q%%nameG@%-%>combstruct/wellfounded/ deriveG6%F`qF_pF`oC$-Feo6)FhnFgo%Gcycle~in~the~dependency~graph~for~sizeGF_p%$a ndGF`q%?derives~an~object~of~this~sizeGO%&falseG>Fcp-%&minusG6$Fcp-%'selectG6%% $hasGFcpF`q[-Feo6(FipFgo%8order~of~types~for~sizeGF_pF[qF`q-Feo6&F@Fgo%;Looking ~for~loops~for~sizeGF4>Fcp-Fep6%F8F4F`o-Feo6(FipFgoFjpF4F[qFcp>F`qFaq@$Feq@$-%? combstruct/wellfounded/derive2G6%F`qF4F`oC$-Feo6)FhnFgo%Jcycle~in~the~dependenc y~graph~for~size~>=GF4F`rF`q%Aderives~an~object~with~such~sizeGOFcr-Feo6(FipFgo F_sF4F[qF`qF9F1F1F16" M6R0 I>combstruct/wellfounded/deriveR6%%"UG%"nG%$valG6"6#%_pCopyright~(c)~1995~by~Ei thne~Murray,~Paul~Zimmermann~and~INRIA.~All~rights~reserved.GF(@'29%&9&6#9$%&fa lseG/F-F.%%trueGY6$Q:please~report~a~bug~in~%16"9!F(F(F(6" M6R0 I?combstruct/wellfounded/derive2R6%%"UG%"nG%$valG6"6#%_pCopyright~(c)~1995~by~E ithne~Murray,~Paul~Zimmermann~and~INRIA.~All~rights~reserved.GF(@%19%&9&6#9$%%t rueGY6$Q:please~report~a~bug~in~%16"9!F(F(F(6" M6R0 IHcombstruct/wellfounded/dependency_graphR6%%$sysG%"nG%$valG6-%"gG%#eqG%$PhiG%$ rhsG%"UG%"VG%"WG%"iG%"jG%"kG%$indG6#%_pCopyright~(c)~1995~by~Eithne~Murray,~Pau l~Zimmermann~and~INRIA.~All~rights~reserved.G6"C&>8$-%&tableGF6>8."""?&8%9$%%tr ueGC$>8'-%#opG6$""#F@@$-%%typeG6$FE%)functionGC%>8(-FG6$F>F@>8&-FG6$""!FE@,/FU. %&UnionGC$>&F96#F=6$-%$seqG6$28)FQ/F`oFE-FG6#-9!6%<#-F]o6$/FQF`oFao9%9&>F=,&F=F >F>F>-FL6$FU.%(productGC%>F`o7#-FG6#FE>8--%%nopsG6#F`o?(8+F>F>FipFB@$/-%(conver tG6$7$-F]o6$&F\p6#&F`o6#8,/F[r;F>,&F^qF>!""F>-F]o6$Fgq/F[r;,&F^qF>F>F>Fip%"+GFX C$>Fin6$2&F`o6#F^qFQ-FG6#-Feo6%<#/FQFjrF[pF\p>F=F^p/FU.%&SubstGC&>F`o-FG6$F>FE> 8*-FG6$FIFE@$/&F\p6#F[tF>C$>Fin6$F_o-FG6#-Feo6%<#FjoF[pF\p>F=F^p@$/&F\pF\qF>C$> Fin6$2F[tFQ-FG6#-Feo6%<#/FQF[tF[pF\p>F=F^p-%'memberG6$FU%Icombstruct/wellfounde d/MultiConstructorsG@$3/-%@combstruct/wellfounded/propertyG6$FUF[p.%"PG-%@combs truct/wellfounded/MultiOneG6#-FG6$;FI-F[qFgpFEC$>Fin6$2FhsFQ-FG6#-Feo6%<#/FQFhs F[pF\p>F=F^p-FL6$FU.%&deltaG@$5/F[pFX0-%Ecombstruct/wellfounded/CoeffOneDeltaG6 #FUFXC$>Fin6$2FfpFQ-FG6#-Feo6%<#/FQFfpF[pF\p>F=F^p-Fbq6$F9%$setGF6F6F66" M6R0 IAcombstruct/wellfounded/has_cycleR6#%"GG6*%&nodesG%'nbpredG%)listsuccG%"iG%#nb G%"jG%&foundG%)epsandatG6#%RCopyright~(c)~1998~by~INRIA.~All~rights~reserved.G6 "C)>8$-%$mapG6$%#opG9$?&8'F4%%trueGC$>&8%6#F;""!>&8&FA%%NULLG>8+-%'selectG6%%%t ypeGF92<$-%*identicalG6#.%%AtomG-FP6#.%(EpsilonG%%nameG?&F;-%&minusG6$F9FHF &F@6#-F86$""#F;,&Fin"""F_oF_o>&FE6#-F86$F_oF;6$F[oFao>8(FB?(F1F_oF_oF10F4<"C(?& F;F40F?FBF1@$F]p-%'RETURNGFA>Fgo,&FgoF_oF_oF_o>&8*6#FgoF;>F4-Fen6$F4<#F;?&8)7#F DF<>&F@6#F\q,&F_qF_o!""F_o7$-F86#-%%map2G6%F8F]oFH-%$seqG6$&FepFA/F;;F_oFgoF1F1 F16" M6R0 IEcombstruct/wellfounded/CoeffOneDeltaR6#%$PhiG6"6#%_pCopyright~(c)~1995~by~Eit hne~Murray,~Paul~Zimmermann~and~INRIA.~All~rights~reserved.GF&@'-%'memberG6$9$< $&%1combstruct/DeltaG6#.%$SetG&F06#.%&CycleG"""5-%%typeG6$F-&F06#%(integerG-F;6 $F-&F06#-%(InCycleGF>@%/-%#opG6#F-F8F8""!@%1-FI6#FHF8F8FKF&F&F&6" M6R0 I@combstruct/wellfounded/MultiOneR6#%(cardresG6"6#%fnCopyright~(c)~1996~Waterlo o~Maple~Inc.~All~rights~reserved.GF&-%&evalbG6#5555/9#""!-%*typematchG6$9$1-%*i denticalG6#.%%cardG%'posintG/F6./F<"""/F6.1FAF8,&F=6#""">8-&F=6#""#>8'&F=6#""$>8.&F=6#FC>8*-%8combstruct/chec kgrammarG6$FGFL>FG&FZFI>FL&FZFN@&4-F;6$FQ-%$setG6#/%%nameG%*procedureGYQhnproce dure~declarations~are~a~set~with~elements~name=procedureF54-F;6$FV-Fao6#/Fdo%*a lgebraicGYQfnmeasure~declarations~are~a~set~with~elements~name=algebraicF5>FV-% &unionG6$FV-%#ifG6%4-%$hasG6$FV.%$nilG<#/F\q""!<">8(-%%map2G6%%#opGFJFQ>8)-Feq6 %FgqFJFV>8+-Fcp6$-Feq6%FgqFJFG<$.%%AtomG.%(EpsilonG?&8$FQ%%trueG>&8&6#-%$lhsG6# Fhr-%6combstruct/parse_procG6'-%$rhsGF`sF^sFcqFiqF]r>8%<#-%$seqG6$/F^sF[s/FhrFQ -%?combstruct/program/wellfoundedG6$FGFgs7&FGFLFgsFVF5F5F5F5 M6R0 I6combstruct/parse_procR6''%"fG%*procedureG'%&fnameG%%nameG%'pnamesG%'mnamesG%' gnamesG6)%"gG%&legalG%%indsG%$errG%'paramsG%$badG%%totoG6"F6C6>8(<#-%#opG6$"""- %%evalG6#9$@$4-%%typeG6$F9-%$setG6#%#::GY6$QPmust~declare~types~of~procedure~ar guments~in~%1F6FB>8)-%&minusG6$-%%map2G6%F<""#F99(@$0FP<"Y6$QXarguments~to~%1~m ust~be~of~types~defined~by~the~grammarF69%@$07#-F<6$""'F?7"Y6$Qindescription~sh ould~not~contain~global~variable~declarations~%1F6Fin>8$-%%subsG6$/.%%NULLG8*-% )procbodyGFA@$-%$hasG6$FeoF\pY6$QJdescription~of~%1~should~not~contain~NULLF6Fi n>8&-%'indetsG6$-FU6%F<""!-Fip6$Feo%(indexedGF`q@$0FgpFenY6$QZdescription~must~ not~contain~list~or~table~references,~%1F6FB>Fgp-%'selectG6$R6#%"xGF66$%)operat orG%&arrowGF6-%&evalbG6#/-%*substringG6$FB;F>F>%"&GF6F6F6-%&unionG6$-FU6%F8%<*%*&functionG%)&statseqG%&&p rocG%'&localG%&&argsG%%&forG%$&ifG%(&expseqG@$0-FR6$FgpF^tFenY6%QHinvalid~algor ithm~description~in~%1,~%2F6FBFjt>8'-%*traperrorG6#-%Acombstruct/parse_proc/sta tementsG6%<#-F<6$""&Feo9&9'@$/F`u%*lasterrorGY6%Q1%1~Happens~in~%2F6F`uFin>Feo- Fbu6#-%:combstruct/parse_proc/forG6%FeoF[vF\v@$/FeoF_vY6%FbvFeoFin>Feo-Fbu6#-%9 combstruct/parse_proc/ifG6&FeoF[vF\vFX@$FjvYF\wFeoF6F6F6F6 M6R0 I:combstruct/parse_proc/forR6%%"gG%'pnamesG%'mnamesG6'%%indsG%%funsG%%varsG%(on evarsG%&exprsG6"F.C->8$-%'indetsG6$9$-%)specfuncG6$%)anythingG%%&forG@$/-%%nops G6#F1""!OF5@$0-%$mapG6$F>F1<#""%YQ8invalid~use~of~for~loopF.>8&-%%map2G6%%#opG" ""F1>8%-F36$FL-F76$F9%*&functionG@$0FS<"C'@$0-FN6%FPFQFS<#.%$OneGYQXillegal~use ~of~function~as~control~variable~in~for-loopF.>8'-FN6%FP""#FS@$50-FE6$F>Fao<#FQ 0-F36$Fao-F76$%%nameG%(&expseqGFaoYQ@improper~use~of~One~in~for-loopF.@$0-%&min usG6$-FN6%FP7$FQF@Fao<#%'&localGFenYQMvariable~inside~One(~~must~be~declared~lo calF.>FL-Fgp6$FLFS@$30FLFen0-FN6%FPF@FLF\qYQKfor-loop~variables~must~be~local~v ariablesF.@$0-%'removeG6%%%typeG-FN6%FPFdoF1%(indexedGFenYQaoinvalid~for-loop:~ ~only~locals~or~procedure~arguments~may~follow~an~inF.>8(-FN6%FP!""F1-%Acombstr uct/parse_proc/statementsG6%Fgr9%9&F5F.F.F.F. M6R0 I9combstruct/parse_proc/ifR6&%"gG%'pnamesG%'mnamesG%'gnamesG60%%indsG%&condsG%" xG%"iG%&typesG%,typematchesG%&exprsG%$badG%*firstargsG%+secondargsG%"tG%%funsG% &sargsG%)specfunsG6"F8CD>8$-%'indetsG6$9$-%)specfuncG6$%)anythingG%$&ifG@$/F;<" OF?>8%<#-%$seqG6$-FM6$-%#opG6$,&8'""#"""FW8&/FU;""!,&-%%iquoG6$-%%nopsG6#FXFVFW !""FW/FXF;@$50-%&minusG6$FJ-F=6$FJ-FA6$FC%*&functionGFG0-Fco6$-%%map2G6%FRFWFJ% 6combstruct/validtestsGFGY6$QZcondition~in~if~must~be~one~of~the~following~func tions~%1F8F`p>8*<#-FM6$-FR6#<$-FM6$-FR6$,$FUFVFX/FU;FWFgn-FR6$F]oFXF^o-%Acombst ruct/parse_proc/statementsG6%Fep9%9&>8(-%'selectG6%%$hasGFJ.%%TypeG>8)-F]r6%F_r FJ.%*TypematchG@$0-Fco6$-%$mapG6$F[o-%&unionG6$-F^p6%FRFVF[r-F^p6%FRFVFcr<#FVFG YQAType/Typematch~takes~2~argumentsF8?&8.<$F[rFcr%%trueG@$-%(hastypeG6$Fjs%%lis tG?&FXFjsF\t@$5-%%typeG6$-FR6$7$FVFWFXFat-Fft6$-FR6$7$FVFVFXFatC%>8,Fht>8-F]u@$ 543-Fft6$FduFat-Fft6$FbuFat0-F[o6#-FR6#Fbu-F[o6#-FR6#FduYQBmisuse~of~lists~in~T ype/TypematchF8>Fbu<#-FM6$-%#ifG6%4-Fft6$FXFatFX-FR6#-FRF\o/FX-F`s6$-F^p6%FRFjt F[rFhw>8+-Fco6$Fbu-F]r6%FftFbu%(indexedG@$0F[xFGY6$QfoFirst~argument~to~Type~mu st~be~a~procedure~argument~or~a~local~variable,~%1F8F[x>Fdu<#-FM6$F\w/FX-F^p6%F RF_uF[r>F[x-Fco6$Fdu9'@$FbxY6$Q^oSecond~argument~to~Type~must~be~a~non-terminal ~from~the~grammar,~%1F8F[x>Fdu<#-FM6$F\w/FX-F^p6%FRF_uFcr>81-F]r6%FftFduFgo@$50 -Fco6$Fdu-F`s6$F]zF`yFG0-Fco6$-F^p6%FRFWF]z<#.FAFGYQ[qSecond~argument~to~Typema tch~must~be~of~the~form~specfunc(~),~or~a~non-terminal~from~the~grammarF8>8/-F^ p6%FR7$FVF]oF]z>F[x-Fco6$Fa[l<$.%%ProdG.%)SequenceG@$FbxY6$Q\ofunction~in~specf unc~must~be~the~constructor~Prod~or~Sequence,~%1F8F[x@$-F_r6$-F]s6$F[o-F^p6%FRF VF]zFWYQ>missing~arguments~in~specfuncF8>Fbu<#-FM6$-FR6$;FW,&FjnFWF]oFWFX/FXFf\ l>80-F]r6%FftFbu<$%#::G%%nameG>F[x-Fco6$FbuFd]l@$FbxY6$Qfnarguments~in~specfunc ~must~be~of~type~name~or~of~type~::~%1F8F[x>F[x-Fco6$-F]r6%FftFd]lFi]l-F`s6$<#. FCF`y@$FbxY6$QTnames~in~specfunc~must~be~defined~in~the~grammar~%1F8F[x>Fd]l-F] r6%FftFd]lFh]l@$0-Fco6$-F^p6%FR7$FWFenFd]l<#%'&localGFGYQTonly~local~variables~ can~be~assigned~using~specfuncF8>F[x-Fco6$-F^p6%FRFVFd]l<$Fi^l-FR6#F`y@$FbxYF\_ lF?F8F8F8F8 M6R0 IAcombstruct/parse_proc/statementsR6%%%statG%'pnamesG%'mnamesG6(%"xG%&exprsG%'f namesG%&namesG%+undeclaredG%%funsG6"F/C+>8%<#-%$seqG6$-%#ifG6%-%%typeG6$8$-%)sp ecfuncG6$%)anythingG%)&statseqG-%#opG6#F=F=/F=9$>8)-%'selectG6%F;F2-F?6$FA%*&fu nctionG>FI-%'indetsG6$FIFM>8&-%%map2G6%FD"""FI@%-%$hasG6$FU%6combstruct/validte stsGY6$QPcannot~use,~%1,~outside~the~condition~of~an~if.F/FhnC&>8(-%&minusG6$FU -%&unionG6$9%9&@$0F^o<"Y6$Q[othe~procedures,~%1,~must~be~have~declared~procedur es~or~measuresF/F^o>F^o-Fco6#-FD6#-%$mapG6%%Bcombstruct/parse_proc/checkparamsG FIFfo@$FhoY6$QVinvalid~or~undeclared~arguments~to~a~function~call~%1F/F^o>8'-FK 6%F;F2%%nameG>F^o-F`o6$F[qFfo@$FhoY6$QSthe~names,~%1,~must~appear~in~measure~de clarationsF/F^o%%trueGF/6#FhnF/F/ M6R0 IBcombstruct/parse_proc/checkparamsR6$%$funG%'mnamesG6#%"rG6"F)C$@'-%%typeG6$9$ %%nameG@%45-%'memberG6$F/9%-F-6$F/%(indexedG>8$<#F/>F<<"4-F56$-%#opG6$"""F/F7>F <-%'removeG6%F-<#-FD6#-FD6$""#F/F:>F<-%&unionG6#-FD6#-%$mapG6%9!FNF7F8$-%&tableG6#7#-%$seqG6$/-%#opG6$"""8(<"/F@9$?&8%9%%%tr ueG?&8&-F=6$7$""#F?FEFGC$@$4-%)assignedG6#&F36#-F=6$FMFIY6$Q6type,~%1,~not~defi nedF0FV>FT-%&unionG6$FT<#-F=6$F?FE?&FEFFFGC&>&8'6#-%$lhsG6#FEFA>8*-%'removeG6%% $hasG-%'indetsG6$-%$rhsGFdo-%)specfuncG6$%)anythingG%*&functionG<&.%%TypeG.%*Ty pematchG.%$OneG.Fap?&8)FfoFG@$/-%&minusG6$<#-F96$&%&&argsG6#8+/Fjq;F?-%%nopsG6# FJ-F\p6$<#-F=6#-F=6$FMF^q%%nameGFA>F_o-Fgn6$F_o<#-F=6$F?F^q>F_o-%*intersectG6$F _o<#-F96$-F=FS/FIFJ-%%evalG6#F`o>FE-%:combstruct/prog/has_cycleGFis@$0FE%&false GY6$QPprocedure,~%1,~does~not~terminate~on~some~inputF0FEFGF0F0F0F0 M6R0 I:combstruct/prog/has_cycleR6#%$depG6+%$relG%%newgG%%rel2G%"UG%"VG%"WG%&graphG% "iG%"xG6#%enCopyright~1995~by~Eithne~Murray,~Paul~Zimmermann~and~INRIAG6"C&>8*< #-%$seqG6$-F86$28,-%#opG6$8+&9$6#F=/FA;"""-%%nopsG6#FB/F=-%$mapG6$F?7#-%(indice sG6#FC?&8$F5%%trueG@$/-F?6$FGFT-F?6$""#FTOFX?(F2FGFGF2FUC&>8%F5?&FTF5FUC%>8'FX> 8(FZ?&8&F5FU@$/Fao-F?6$FGFcoC$>8)-F?6$FfnFco@%/F_oFjoOF_o>F[o-%&unionG6$F[o<#2F _oFjo@$/F[oF5[>F5F[o%&falseGF2F2F2F2 M6R0 I6combstruct/validtests<$%%TypeG%*TypematchG6" M6R0 I3combstruct/compileR6$%$SysG%$typG6$%$sysG%$valG6$%)rememberG%fnCopyright~(c)~ Eithne~Murray~and~INRIA.~All~rights~reserved.G6"C)-%'forgetG6$%2combstruct/Name OfG/%-subfunctionsG%&falseG>%8combstruct/user_spec_NFG6$9$9%>8$-%8combstruct/st andardformGF8>8%-%Gcombstruct/wellfounded/find_valuationsG6#F<-%8combstruct/cou nt_tablesG6$F&&%0combstruct/DrawG6#%8combst ruct/user_spec_NFG6#.%(EpsilonGRF-F-6$%)operatorG%&arrowGF-F6F-F-F->&F16#.%%Ato mGRF-F-F9F-F?F-F-F-?&8$9$%%trueGC$@%-%'memberG6$-%$lhsG6#FC<#-%$seqG6$-FL6#8&/F T%8combstruct/user_sys_NF1G>8%.%)toplevelG>FX.%&innerG>&F16#FK-%%subsG6$7&/%combstruct/placeholder/count2G&%2combstruct/Count2GF3/%=combstruct/placeh older/draw2G&%1combstruct/Draw2GF3-&%4combstruct/makedrawG6#FX6$FC9%F-6%F2F4FVF -6" M6R0 I8combstruct/count_tablesR6$%$sysG%$valG6#%#eqG6#%fnCopyright~(c)~Eithne~Murray ~and~INRIA.~All~rights~reserved.G6"C&>&&%1combstruct/CountG6#%8combstruct/user_ spec_NFG6#.%(EpsilonG-%=combstruct/makecount/epsilonGF+>&F/6#.%%AtomG-%:combstr uct/makecount/atomGF+?&8$9$%%trueG>&F/6#-%$lhsG6#F@-%%subsG6$7$/%=combstruct/pl aceholder/countGF//%>combstruct/placeholder/count2G&%2combstruct/Count2GF1-%5co mbstruct/makecountG6$F@9%@$%3combstruct/usegfunG-%;combstruct/count/with_gfunG6 #FAF+6%F0F2FXF+6" M6R0 I5combstruct/makecountR6$%#eqG%$valG6$%$PhiG%#rsG6#%_pCopyright~(c)~1995~by~Eit hne~Murray,~Paul~Zimmermann~and~INRIA.~All~rights~reserved.G6"C$>8%-%#opG6$""#9 $@*-%%typeG6$F/.%(epsilonG-%=combstruct/makecount/epsilonGF,-F76$F/.%%atomG-%:c ombstruct/makecount/atomGF,-F76$F/.%,nonterminalG-%:combstruct/makecount/nameG6 #F/-F76$F/%)functionGC$>8$-F16$""!F/@1/FO.%&UnionG-%;combstruct/makecount/Union G6#7#-F1FI-F76$FO.%(productG-&%=combstruct/makecount/ProductG6#&%8combstruct/us er_spec_NFG6#F36%-F16$"""F/-F16$F3F/9%-F76$FO.%&substG-%;combstruct/makecount/S ubstG6$FenFgo-F76$FO.%&deltaG-%;combstruct/makecount/DeltaG6$Fen-F16#FO/FO.%/co mbstruct/IntG-%9combstruct/makecount/IntG6#Fen/FO.%1combstruct/ThetaG-%;combstr uct/makecount/ThetaGF]q/FO.%0combstruct/PSetG-%:combstruct/makecount/PSetGF]qY6 $Q7unknown~constructor~%16"FOF,6#F_oF,6" M6R0 I=combstruct/makecount/epsilonR6"F$6#%_pCopyright~(c)~1995~by~Eithne~Murray,~Pa ul~Zimmermann~and~INRIA.~All~rights~reserved.GF$R6#%'gArg_1GF$6#%)rememberGF$@% /9$""!"""F/F$F$F$F$F$F$6" M6R0 I:combstruct/makecount/atomR6"F$6#%_pCopyright~(c)~1995~by~Eithne~Murray,~Paul~ Zimmermann~and~INRIA.~All~rights~reserved.GF$R6#%'gArg_1GF$6#%)rememberGF$@%/9$ """F/""!F$F$F$F$F$F$6" M6R0 I:combstruct/makecount/nameR6#%"AG6"6#%_pCopyright~(c)~1995~by~Eithne~Murray,~P aul~Zimmermann~and~INRIA.~All~rights~reserved.GF&-%%subsG6$/%@combstruct/placeh older/nontermAG9$R6#%'gArg_1GF&6#%)rememberGF&-&%=combstruct/placeholder/countG 6#F-6#F.F&F&F&F&F&F&6" M6R0 I;combstruct/makecount/UnionR6#%"lG6$%$defG%"xG6#%_pCopyright~(c)~1995~by~Eithn e~Murray,~Paul~Zimmermann~and~INRIA.~All~rights~reserved.G6"C$>8$-%(convertG6$7 #-%$seqG6$-&%=combstruct/placeholder/countG6#8%6#%'gArg_1G/F:9$%"+G-%%subsG6$/% 8%-%#opG6#%Mcombstruct/count/Product/template/unlabel ledG-%%subsG6'/%@combstruct/placeholder/nontermAGF7/%@combstruct/placeholder/no ntermBGF=/%FU-FW6#%Kcombstruct/count/Product/template/labelledG-F en6$7&FgnFinF[oF_oFboF2F2F2F$ M6R0 I8$-%%subsG6$/9&%'gArg_1G9'-F26$7%/%F AF8%FJ@*/F1.%$SetG>8$FJ/F1.%&CycleG>FT.-&%*numt heoryG6#%$phiG6#%'gLoc_1G-F/6$F1-FR6#F2C$>FTFJ>FN-%#opG6#F1-F/6$F1-FWF^oC$>FTFY >FNFbo-F46$7%/%FNFC?(FTFapFJF8%%trueG@$/-FA6$F8FTFC>FN,& FNFJ*&F_pFJ-FE6#*&F8FJFTFKFJFJFNF,F,F,F,F,F,6" M6R0 I9combstruct/makecount/IntR6#%"AG6"6#%_pCopyright~(c)~1995~by~Eithne~Murray,~Pa ul~Zimmermann~and~INRIA.~All~rights~reserved.GF&-%%subsG6$/%@combstruct/placeho lder/nontermAG9$R6#%'gArg_1GF&6#%)rememberGF&@%/F.""!F6*&-&%=combstruct/placeho lder/countG6#F-6#F."""F.!""F&F&F&F&F&F&6" M6R0 I;combstruct/makecount/ThetaR6#%"AG6"6#%_pCopyright~(c)~1995~by~Eithne~Murray,~ Paul~Zimmermann~and~INRIA.~All~rights~reserved.GF&-%%subsG6$/%@combstruct/place holder/nontermAG9$R6#%'gArg_1GF&6#%)rememberGF&*&F."""-&%=combstruct/placeholde r/countG6#F-6#F.F5F&F&F&F&F&F&6" M6R0 I;combstruct/makecount/SubstR6%%"AG%"BG%$valG6#%%flagG6#%_pCopyright~(c)~1995~b y~Eithne~Murray,~Paul~Zimmermann~and~INRIA.~All~rights~reserved.G6"C%>8$-%&eval bG6#/&%8combstruct/user_spec_NFG6#""#.%)labelledG>&&%2combstruct/Count2G6#F56#9 $-%%subsG6$7&/%@combstruct/placeholder/nontermAGF@/%Acombstruct/placeholder/lab elflagGF//%=combstruct/placeholder/countG&%1combstruct/CountGF>/%>combstruct/pl aceholder/count2GF<-%#opG6#%@combstruct/count/power/templateG-FB6$7&FE/%@combst ruct/placeholder/nontermBG9%/%8$-%$rhsG6#9$@*/F4.%(EpsilonG@%-%%typeG6$-%$lhsGF7.%)uniontagG>8%-%%subs G6$/%@combstruct/placeholder/nontermAGFARF1F1F1F1FKF1F1F1>FFRF1F16$%)operatorG% &arrowGF1F;F1F1F1/F4.%%AtomG>FFRF1F1FOF1FSF1F1F1-F?6$F4.%,nonterminalG>FF-FH6$/ FKF4R6$%'gArg_1G%'gArg_2GF1F1F1-&%FF-%=combstruct/makedraw/functionG6$F4Fbo-%%evalG6#FFF1F1F1% )toplevelGRF)6(F-F.%%origG%,newfunctionG%*printtypeG%(canfuncGF/F1C(>F4F5@*F:>F F-FH6$FJRF1F1F1F1FKF1F1F1FR>FF-FH6$FJRF1F1F1F1FKF1F1F1FW>FF-FH6$FhnRFjnF1F1F1F] oF1F1F1Fco>FFFgo>8'FF>8&-%#opG6#-%'indetsG6$%8combstruct/user_sys_NF1G/-%*ident icalG6#FA%)anythingG@$-F?6$-F66#FgqFeoC%@$5F:FR>FF-FH6$FhnRF1F1F1F1FKF1F1F1>8(- Fiq6$""!Fgr@(-F?6$Fas.%(usecardGC$@%3/&%8combstruct/user_spec_NFG6#""#.%+unlabe lledG-%'memberG6$Fas7%.%$SetG.%&CycleG.%)PowerSetG>8)%@combstruct/draw/canonica l_unlabG>F_uRF1F1FOF19"F1F1F1>Feq-FH6$7%/%Feq-FH6$7$FhuFjuRFjnF1F1F 1FdvF1F1F1-F?6$Fas.%&deltaG>Feq-FH6$FjuRFjnF1F1F17#FfvF1F1F1-F[p6#FeqF16#F^rF1F $ M6R0 I?combstruct/makecount/PSet/cardR6'%"pG%"nG%"BG%$sysG%$typG6%%&totalG%"kG%"iG6$ %)rememberG%fnCopyright~(c)~1996~Waterloo~Maple~Inc.~All~rights~reserved.G6"@)2 9%9$""!5/F5F6/F4F6@%3F8F9"""F6/F5F<-&&%1combstruct/CountG6$9'9(6#9&6#F4C&>8$F6? (8%FFJ,&FJF<*()FQ, &FLFFJ,&FJF<* &)FQ,&F5F8$-%#opG6$""!9$@0/F..%&UnionG-%:comb struct/makedraw/UnionG6#-F06#F3/F..%%ProdG-% 8&7#F+>8%9%?(8$"""F?F$%%trueGC$>8'-&%=combstruct/placeholder/countG6#-%#opG6$F> F86#9$@%2F;FC[>F;,&F;F?FC!""-&%8$R6$%'gA rg_1G%'gArg_2GF-6#%fnCopyright~(c)~1996~Waterloo~Maple~Inc.~All~rights~reserved .GF-6$%@combstruct/placeholder/nontermAG-&%F?RFAF-FDF-6$FG -FI6$FNFTF-F-F--F16$F3.%(epsilonG>F?RFAF-FDF-6$FG-FI6$F3FTF-F-F--F16$FTF4@%F7>F ?RFAF-FDF-6$-&FJ6#FGFMFLF-F-F->F?RFAF-FDF-6$-FdoFYFLF-F-F--F16$FTFfn>F?RFAF-FDF -6$-FdoF\oFLF-F-F-C$@%/%2combstruct/methodG.%*optimizedGC$>8%-%5combstruct/heur isticG6&F9F3FT"#I-%)userinfoG6)F;%+combstructG%3best~heuristic~forGF3FT.%#isGFh p>FhpFcp>F?-%#opG6#-%$catG6&%Bcombstruct/draw/Product/template/GF8%"/GFhp-%%sub sG6$7&/FGF3/FLFT/%8%"""@ +/9%.%$SetG>8$F0/F3.%&CycleG>F7.-&%*numtheoryG6#%$phiG6#%'gLoc_1G-%%typeG6$F3-F 56#%(integerGC$>F7F0>F/-%#opG6#F3-FE6$F3-F:FHC$>F7F<>F/FMO-%%subsG6$7$/%@combst ruct/placeholder/nontermAG9$/%9combstruct/placeholder/kGF3R6$%'gArg_1G%'gArg_2G FBF,F,-%$seqG6$-FN6#F7/F77#-%"$G6$7#-&%F/F3?(F7Ff pF0Fgn%%trueG@$/-%$modG6$FgnF7""!C$>8&*&FdpF0-&%=combstruct/placeholder/countGF \p6#*&FgnF0F7F_pF0@%2F/Fiq[>F/,&F/F0FiqF_p@%/FdpF0>8'F/>Fhr-%%iquoG6$F/Fdp-F_o6 $-FN6#8(/Fas7#-Ffo6$7#-Fjo6$F_rFhrF7F,F,F,F,F,F,6" M6R0 I:combstruct/makedraw/SubstR6$%"AG%"BG6#%%flagG6#%_pCopyright~(c)~1995~by~Eithn e~Murray,~Paul~Zimmermann~and~INRIA.~All~rights~reserved.G6"C%>8$-%&evalbG6#/&% 8combstruct/user_spec_NFG6#""#.%)labelledG>&&%1combstruct/Draw2G6#F46#9$-%%subs G6$7(/%@combstruct/placeholder/nontermAGF?/%Acombstruct/placeholder/labelflagGF ./%=combstruct/placeholder/countG&%1combstruct/CountGF=/%>combstruct/placeholde r/count2G&%2combstruct/Count2GF=/%combstruct/draw/Subst/doSubstR6$%"sG%"xG6"6#%_pCopyright~(c)~1995~by~Eithne~M urray,~Paul~Zimmermann~and~INRIA.~All~rights~reserved.GF'-%#opG6$7$"""F.-%Bcomb struct/draw/Subst/doSubst_auxG6%7#9%F.9$F'F'F'6" M6R0 IBcombstruct/draw/Subst/doSubst_auxR6%%"xG%)startindG%)subslistG6&%%newxG%"iG%" rG%$indG6#%_pCopyright~(c)~1995~by~Eithne~Murray,~Paul~Zimmermann~and~INRIA.~Al l~rights~reserved.G6"C%>8'9%?(8%"""F6-%%nopsG6#9$%%trueG@'-%%typeG6$&F:6#F5.%(e psilonG>&8$FAF@-F>6$F@%%nameGC$>FE&9&6#F2>F2,&F2F6F6F6C%>8&-9!6%7#-%#opG6#F@F2F M>FE--FY6$""!F@6#-FY6#&FS6#F6>F2,&F2F6&FS6#""#F67$7#-%$seqG6$FE/F5;F6F7,&F2F6F3 !""F/F/F/6" M6R0 I:combstruct/makecount/PSetR6$%"TG%"BG6#%%restG6#%_pCopyright~(c)~1995~by~Eithn e~Murray,~Paul~Zimmermann~and~INRIA.~All~rights~reserved.G6"C$@%/9#""$>8$&9"6#F 0>F2%&falseG-%%subsG6$7%/%@combstruct/placeholder/nontermBG9%/%;combstruct/plac eholder/sysG%8combstruct/user_spec_NFG/%8*-%?c ombstruct/makecount/PSet/cardG6&Fgn9$F=F@C&>8)F7@$5/FCF7-FT6$FC1FfnFW@%30FCF72F ^oFgn>Fjn""!C'?(F2"""F`pF^o%%trueG>&8&6#F2-%$addG6$**)!"",&8%F`pF`pF`pF`pF2F`pF ]qF[q-&%=combstruct/placeholder/countG6#F=6#*&F2F`pF]qF[qF`p/F]q-&%*numtheoryG6 #%)divisorsGFep>&8'6#F]pF`p?(F2F`pF`pF^oFap>&F\rFep*&-Fgp6$*&&F\r6#F]qF`p&Fdp6# ,&F2F`pF]qF[qF`p/F]q;F]p,&F2F`pF[qF`pF`pF2F[q>Fjn&F\r6#F^o@$3Fjo2F]pFgn>FaoFap@ $5-FT6$FC1FW'Fgn%'posintGFaoC&@$FaoC$>Fgn,&FgnF`pF[qF`p>8+Fjn>FjnF]p?(F2FgnF[qF ]pFap>Fjn,&FjnF`p-F\o6&F2F^oF=F@F`p@$Fao>Fjn,&FatF`pFjnF[qFjnF+F+F+F+6#FAF+6" M6R0 I9combstruct/makedraw/PSetR6#%"TG6"6#%_pCopyright~(c)~1995~by~Eithne~Murray,~Pa ul~Zimmermann~and~INRIA.~All~rights~reserved.GF&-%%subsG6$/%@combstruct/placeho lder/nontermAG9$R6$%'gArg_1G%'gArg_2G6$%'gLoc_1G%'gLoc_2GF&F&C&>8$-&%=combstruc t/placeholder/countG6#F-6#F.>8%7#-&%F?F@FPF&F&F&F&F& F&6" M6R0 ILcombstruct/draw/Product/template/unlabelledR6$%'gArg_1G%'gArg_2G6'%'gLoc_1G%' gLoc_2G%'gLoc_3G%'gLoc_4G%'gLoc_5G6#%_pCopyright~(c)~1995~by~Eithne~Murray,~Pau l~Zimmermann~and~INRIA.~All~rights~reserved.G6"C'>8%9%@%/%8(%F9,&9$F7%8&*&-&%=combstruct/placeholder/countG6#%@combstruct/ placeholder/nontermAG6#FAF7-&FI6#%@combstruct/placeholder/nontermBG6#,&F=F7FAF? F7@%2F2FE[>F2,&F2F7FEF?>FE-%%iquoG6%F2FG.8'6$-&%8%9%>8("""@%/ %8)""!>F=9$?(8$F=F;F1%%trueGC%>8**(F7F8-&%=comb struct/placeholder/countG6#%@combstruct/placeholder/nontermAG6#FBF8-&FJ6#%@comb struct/placeholder/nontermBG6#,&F@F8FB!""F8@%2F4FF[>F4,&F4F8FFFT>F7*&F7F8)*&,*F @F8FBFTF;#FT""##F8FjnF8F8,(FBF8F[oF8F;F[oFTF;F8>8&-%%iquoG6%F4F7.F4>F^o-F`o6%F^ oFH.8'6$-&%8$F<@%%Acombstruct/placeholder /labelflagGC$?(8(F4F4,(F;F4F3!""F4F4%%trueGC%>8'-%)binomialG6$,&F;F4FFF4,&FDF4F FF4>8&*(FJF4-&%=combstruct/placeholder/countGF86#FDF4-&%>combstruct/placeholder /count2GF86$,&F;F4FDFF,&F3F4FFF4F4@%2F?FQ[>F?,&F?F4FQFF>8%-%%iquoG6%-F_o6$F?FJF S.F?C$?(FDF4F4FEFGC$>FQ*&FSF4FWF4@%FhnFin>F?F[o>F]o-F_o6%F?FSFco6$-F66$FDF?-&%= combstruct/placeholder/draw2GF86%FenFfnF]oF0F0F06" M6R0 I?combstruct/draw/Subst/templateR6$%'gArg_1G%'gArg_2G6'%'gLoc_1G%'gLoc_2G%'gLoc _3G%'gLoc_4G%'gLoc_5G6#%_pCopyright~(c)~1995~by~Eithne~Murray,~Paul~Zimmermann~ and~INRIA.~All~rights~reserved.G6"C'>8%9%>8'9$?(8$"""F9,&F6F9F5F9%%trueGC$>8(*& -&%=combstruct/placeholder/countG6#%@combstruct/placeholder/nontermBG6#F8F9-&%> combstruct/placeholder/count2G6#%@combstruct/placeholder/nontermAG6$F6F8F9@%2F2 F>[>F2,&F2F9F>!"">8&-%%iquoG6%F2F@.F2-%>combstruct/draw/Subst/doSubstG6$7#-&%=c ombstruct/placeholder/draw2GFI6%F6F8FS-&%8%""!>8&-%)binomialG6$9$%F1,& F1F<*(F4F<-&%=combstruct/placeholder/countG6#%@combstruct/placeholder/nontermAG 6#F;F<-&FG6#%@combstruct/placeholder/nontermBG6#,&F8FF4*(F4F8%""!?(8$%F0,&F0F5*&-&%=combstruct/placehold er/countG6#%@combstruct/placeholder/nontermAG6#F3F5-&F@6#%@combstruct/placehold er/nontermBG6#,&F7F5F3F9F5F5F0F-F-F-6" M6R0 I@combstruct/count/power/templateR6$%'gArg_1G%'gArg_2G6$%'gLoc_1G%'gLoc_2G6%%)r ememberG%'systemG%_pCopyright~(c)~1995~by~Eithne~Murray,~Paul~Zimmermann~and~IN RIA.~All~rights~reserved.G6"@%/9%"""-&%=combstruct/placeholder/countG6#%@combst ruct/placeholder/nontermAG6#9$C$@%%Acombstruct/placeholder/labelflagGC$>8$""!?( 8%F2F2,(F9F2F1!""F2F2%%trueG>F?,&F?F2*(-%)binomialG6$,&F9F2FDF2,&FBF2FDF2F2-F46 #FBF2-&%>combstruct/placeholder/count2GF66$,&F9F2FBFD,&F1F2FDF2F2F2>F?-%$addG6$ *&FNF2FPF2/FB;F2FCF?F.F.F.6" M6R0 I@combstruct/count/Subst/templateR6#%'gArg_1G6$%'gLoc_1G%'gLoc_2G6%%)rememberG% 'systemG%_pCopyright~(c)~1995~by~Eithne~Murray,~Paul~Zimmermann~and~INRIA.~All~ rights~reserved.G6"C%>8%9$@$2"""%F0-%$minG6$F0-%% iquoG6$F1F5-%$addG6$*&-&%=combstruct/placeholder/countG6#%@combstruct/placehold er/nontermBG6#8$F4-&%>combstruct/placeholder/count2G6#%@combstruct/placeholder/ nontermAG6$F1FGF4/FG;%8%9%?( 8$""!"""F0%%trueGC(>8(F6>8)*&-&%=combstruct/placeholder/countG6#%@combstruct/pl aceholder/nontermAG6#F[>F3,&F3F8F>FM>F<,&FLF8F6FM>F>F?@%FOFP>F3FR>8&-%%iquoG6%F3F@.8'6$-&%8%9%>8(""">8+-%%evalG6#&%=combstruct/placeholder/countG6#%@c ombstruct/placeholder/nontermAG>8,-F>6#&FA6#%@combstruct/placeholder/nontermBG? (8$""!F:F3%%trueGC)>8)FL>8**(F9F:-F<6#FQF:-FE6#,&9$F:FQ!""F:@%2F6FS[>F6,&F6F:FS Fen>FQ,&FZF:FLFen>FSFT@%FgnFhn>F6Fjn>F9*(F9F:F\oF:,&FLF:F:F:Fen>8&-%%iquoG6$F6F 9>Fdo-Ffo6%Fdo-F@FV.8'6$-&%8%9%@%/!""""">8(%F9,&9$F7%8&*&-&%=combstruct/placeholder/countG6#%@combstruct/placeholder/nonter mAG6#F@F7-&FH6#%@combstruct/placeholder/nontermBG6#,&F=F7F@F6F7@%2F2FD[>F2,&F2F 7FDF6>FD-%%iquoG6%F2FF.8'6$-&%8%9% >8("""@%/F8F8>8)""!>F<9$?(8$F8**(F7F8-&%=combstruct/placeholder/ countG6#%@combstruct/placeholder/nontermAG6#FAF8-&FI6#%@combstruct/placeholder/ nontermBG6#,&F?F8FA!""F8@%2F4FE[>F4,&F4F8FEFS>F7*(F7F8FRF8,&FAF8F8F8FS>8&-%%iqu oG6%F4F7.F4>Fgn-Fin6%FgnFG.8'6$-&%8%9 %>8("""@%/!""F8>8)""!>F=9$?(8$F=F;F1%%trueGC%>8**(F7F8-&%=combstruct/placeholde r/countG6#%@combstruct/placeholder/nontermAG6#FBF8-&FJ6#%@combstruct/placeholde r/nontermBG6#,&F@F8FBF;F8@%2F4FF[>F4,&F4F8FFF;>F7*(F7F8,(F@F8FBF;F8F8F;FBF8>8&- %%iquoG6%F4F7.F4>Fgn-Fin6%FgnFH.8'6$-&%8$-%@combstruct/count/holonomic_partG6#9$@$0F57"C$>8%-%;comb struct/count/make_eqnsG6#F5?&8&F?F2C$>8'-%#opG6#-%&minusG6$-%'indetsG6$FD%%name G<#.%"zG@$4-%(hastypeG6$F9/-%*identicalG6#FGFQ>&&%1combstruct/CountG6#%8combstr uct/user_spec_NFGFgn-%:combstruct/count/gfunprocG6#FDF-6$F[oF]oF-6" M6R0 I:combstruct/gfun_installedR6"6#%"lG6#%_pCopyright~(c)~1995~by~Eithne~Murray,~P aul~Zimmermann~and~INRIA.~All~rights~reserved.GF$@%4-%%typeG6$&%%gfunG6#%,diffe qtorecG%*procedureGYQaoThe~gfun~package~is~not~installed.~See~combstruct[option s]~for~details6"C$@%43-%)assignedG6#%-gfun/versionG1$"#A!""F=YQjnThe~gfun~packa ge~version~is~too~old.~~Need~version~2.2~or~laterF5C$>8$7%&F/6#%.algeqtodiffeqG F.&F/6#%)invborelG@$-%$hasG6$-%$mapG6%F,FFF2%&falseGYQaoThe~gfun~package~is~inc ompatible.~~See~combstruct[options]~for~detailsF5%%trueGF$F$F$F5 M6R0 I@combstruct/count/holonomic_partR6#%$SysG6-%"iG%&fDoneG%$sysG%+closed_sysG%&ad dedG%%csysG%)nontermsG%#eqG%"xG%$ntsG%"jG6#%fnCopyright~(c)~Eithne~Murray~and~I NRIA.~All~rights~reserved.G6"C$>8&-%'selectG6%%%typeG9$<$/%%nameG-%)specfuncG6$ F?<$.%&UnionG.%%ProdG/F?F?@%0F7<"C)>8*<%.%(EpsilonG.%%AtomG-%$seqG6$-%$lhsG6#8, /FZF7>8%%&falseG?(F4"""FjnF44FgnC$>Fgn%%trueG?&8+F7F^oC$@%-F;6$-%$rhsG6#F`o%)fu nctionG>8-<#-%#opG6#Feo>Fjo<#Feo@$4-%&evalbG6#/-%*intersectG6$FNFjoFjoC%>F7-%&m inusG6$F7<#F`o>FN-F]q6$FN<#-FXFgo>FgnFhn>8$""!>FgnFhn?(F4FjnFjnF4F[oC(>8'<#&F76 #Fjn>8(F^r>Fgq,&FgqFjnFjnFjn?(F4FjnFjnF430FbrFK0F]rF7C$>Fbr-F]q6$-F96%%$hasGF7- F]q6$-%'indetsG6$FbrF?<$FPFRF]r>F]r-%&unionG6$F]rFbr>&8)6#FgqF]r@%/F]rF7>FgnF^o >F7-F]q6$F7F]r7#-FU6$&F\t6#8./Fit;FjnFgq7"F4F4F46" M6R0 I;combstruct/count/make_eqnsR6#%%hsysG6/%%eqnsG%%indsG%%varsG%%newsG%"sG%"rG%"i G%"xG%"nG%"jG%%neqsG%"bG%$varG6#%fnCopyright~(c)~Eithne~Murray~and~INRIA.~All~r ights~reserved.G6"C&>8$-%$mapG6&&%+combstructG6#%'gfeqnsG9$.%)labelledG.%"zG>8* """?&8(F9%%trueGC,>8%-%'indetsG6$FJ%)functionG>8&7#-%$seqG6$-%#opG6$""!8+/FgnFN >8'-%%subsG6$<#-FW6$/FgnFYFhnFJ>Fjn7#-FW6$,&-%$lhsG6#FgnFH-%$rhsGFio!""/FgnFjn> 8/-&%(grobnerG6#%'gbasisG6%FjnFT.%%plexG>8.-%%nopsG6#F_p>F_p-%(convertG6$F_p%&t ableG?(8,FipF\pF64-F_q6$<#-FW6$-%&evalbG6#1-F[q6#-FP6$&F_p6#8-%%nameG""#/Fdr;FH Fip%$andGC$>80-FZ6#-%&minusG6$-FP6$&F_p6#FcqFer<#FD?(FdrFHFH,&FcqFHF\pFHFK@$-%$ hasG6$FbrF\s>Fbr-%)primpartG6#-%*resultantG6%FbrFdsF\s>&8)6#FG-F_q6$F_p%%listG> FG,&FGFHFHFH-F;6$FZ-F_q6$FftFjtF6F6F66" M6R0 I:combstruct/count/gfunprocR6#%$geqG6)%(nontermG%$deqG%*initcondsG%$recG%"aG%"n G%"iG6#%fnCopyright~(c)~Eithne~Murray~and~INRIA.~All~rights~reserved.G6"C*>8$-% #opG6#-%&minusG6$-%'indetsG6$9$%%nameG<#.%"zG>8%-&%%gfunG6#%.algeqtodiffeqG6$F= -F36#F@@$-%%typeG6$FC%$setG>FC-F56#-%'removeG6%FNFC%)equationG>8&-%$seqG6$/---% #@@G6$%"DG8*6#F36#""!*&-%*factorialG6#F^o"""-&&%1combstruct/CountG6#%8combstruc t/user_spec_NFGF_oFeoFfo/F^o;Fao,&-%'degreeG6$F=F3Ffo!""Ffo>8'-&FF6#%,diffeqtor ecG6%<$FCFYFJ-8(6#8)@$/&F\p6#""#.%)labelledG>Fep-&FF6#%)invborelG6$FepF\q>FY-%' selectG6%R6$%"xGF,F06$%)operatorG%&arrowGF04-%$hasG6$F:9%F0F0F0FepF_q-&%)LREtoo lsG6#%)REtoprocG6&-F86$FepFYF\qFY.%)rememberGF06$FjoF\pF06" M6R0 I1combstruct/countR6"6#%"rG6#%^oCopyright~(c)~1995~by~Eithne~Murray~and~INRIA.~ All~rights~reserved.GF$C%@$0%7combstruct/InitializedG%%trueG-%0combstruct/initG F$>8$-%6combstruct/parse_argsG6#9"@%-%%typeG6$&F16#"""-%*identicalG6#.%(grammar G-%8combstruct/countgrammarG6$&F16#""#&F16#""$-%8combstruct/countspecialGFDF$F$ F$6" M6R0 I8combstruct/countgrammarR6$%%specG%%sizeG6%%"yG%$sysG%$typG6#%_pCopyright~(c)~ 1995~by~Eithne~Murray,~Paul~Zimmermann~and~INRIA.~All~rights~reserved.G6"C'>8$& 9$6#""">8%&F26#""#>8&&F26#""$-%3combstruct/compileG6$F6F;-&&%1combstruct/CountG FA6#F06#9%F-F-F-6" M6R0 I8combstruct/countspecialR6$%*structureG%"nG6&%#fnG%(arglistG%'fnnameG%$errG6#% _pCopyright~(c)~1995~by~Eithne~Murray,~Paul~Zimmermann~and~INRIA.~All~rights~re served.G6"C&@'3-%%typeG6$9$.%)functionG-F36$-%#opG6$""!F5.%%nameGC$>8&F:>8%-F;6 #F5-F36$F5F>C$>FBF5>FD%%NULLGY6$Q:invalid~structure~name~%16"F5>8$(%2combstruct /count/GFB@$4-F36$FR%*procedureGZ%>8'-%(readlibG6#FRF.Y6$Q7undefined~structure~ %1FPFB-FR6$9%FDF.F.F.FP M6R0 I0combstruct/drawR6"6#%"rG6#%^oCopyright~(c)~1995~by~Eithne~Murray~and~INRIA.~A ll~rights~reserved.GF$C%@$0%7combstruct/InitializedG%%trueG-%0combstruct/initGF $>8$-%6combstruct/parse_argsG6#9"@%-%%typeG6$&F16#"""-%*identicalG6#.%(grammarG -%7combstruct/drawgrammarG6$&F16#""#&F16#""$-%7combstruct/drawspecialGFDF$F$F$6 " M6R0 I7combstruct/drawspecialR6$%*structureG%"nG6&%#fnG%(arglistG%$errG%'fnnameG6#%_ pCopyright~(c)~1995~by~Eithne~Murray,~Paul~Zimmermann~and~INRIA.~All~rights~res erved.G6"C&@'3-%%typeG6$9$.%)functionG-F36$-%#opG6$""!F5.%%nameGC$>8'F:>8%-F;6# F5-F36$F5F>C$>FBF5>FD%%NULLGY6$Q:invalid~structure~name~%16"F5>8$(%1combstruct/ draw/GFB@$4-F36$FR.%*procedureGC$>8&-%*traperrorG6#-%(readlibG6#FR@$/Fgn%*laste rrorGY6$Q7undefined~structure~%1FPFB-FR6$9%FDF.F.F.FP M6R0 I7combstruct/drawgrammarR6$%%specG%%sizeG6#%$objG6#%_pCopyright~(c)~1995~by~Eit hne~Murray,~Paul~Zimmermann~and~INRIA.~All~rights~reserved.G6"C$Z%>8$-%;combstr uct/draw/tagged_objG6$9$9%F+YF+-%8$-&%)combinatG6#%)randpermG6#F/-F26$7$"""FA-%:combstruct/draw/l abel_auxG6&F4FAF79&F,F,F,6" M6R0 I:combstruct/draw/label_auxR6&%"xG%)startindG%'labelsG%)atomlistG6&%$indG%%newx G%"rG%"iG6#%fnCopyright~(c)~1996~Waterloo~Maple~Inc.~All~rights~reserved.G6"C%> 8$9%?(8'"""F7-%%nopsG6#9$%%trueG@'-%'memberG6$&F;6#F69'C$>&8%FB&FA6#&9&6#F3>F3, &F3F7F7F74-%%typeG6$FA%)functionG>FFFAC%>8&-9!6&7#-%#opG6#FAF3FKFC>FF--Fgn6$""! FA6#-Fgn6#&FW6#F7>F3,&F3F7&FW6#""#F77$7#-%$seqG6$FF/F6;F7F8,&F3F7F4!""F0F0F06" M6R0 I5combstruct/draw/RandR6#%"nG6"6$%)rememberG%fnCopyright~(c)~1996~Waterloo~Mapl e~Inc.~All~rights~reserved.GF&-%%randG6#9$F&F&F&6" M6R0 I=combstruct/draw/canonicalizeR6$%%exprG%$typG6$%"rG%"xG6#%fnCopyright~(c)~1996 ~Waterloo~Maple~Inc.~All~rights~reserved.G6"@%-%%typeG6$9$<'%)relationG%(intege rG%%nameG-%*identicalG6#.%(EpsilonG-F76#.%%AtomGF1C%@%-%(hastypeG6$7#-%#opG6#F1 -%)specfuncG6$%)anythingG<%.%$SetG.%)PowerSetG.%&CycleG>8$7#-%$seqG6$-9!6$8%9%/ FfnF1>FTFD@&/-FF6$""!F1FQ@%/Fgn.%)labelledG>FT7#-%:combstruct/draw/canCycLabG6# -FF6#FT>FT7#-%FT-%%sortG6 $FT.%(addressG@%-F/6$F1%%listGFT-F\oFgoF,F,F,6" M6R0 I:combstruct/draw/canCycLabR6"6&%"iG%$elsG%)minlabelG%"xG6#%fnCopyright~(c)~199 6~Waterloo~Maple~Inc.~All~rights~reserved.GF$C&>8%7#9">8&-%$minG6#-%$seqG6$-%#o pG6#8'/F<-%'indetsG6$F.%(indexedG?(8$"""FD-%%nopsG6#F.4-%$hasG6$&F.6#FCF2F$6$-F :6#&F.6#;FC!""-F:6#&F.6#;FD-%$maxG6$""!,&FCFDFTFDF$F$F$6" M6R0 I8$7#F:>8%-%%sortG6$ F>.%(addressG@%0&FA6#F;&FA6#""#-%'memberG6%FIF>.8)C*>8&FI?(8+F;F;-F76#F>%%trueG @$/&F>6#FWFUC$>&8'6#8,FW>F^o,&F^oF;F;F;>8(-%(convertG6$F\o%%listG>F\o.F\o>8/R6$ %"LGF-6#F.F2F$C%>F>-%%modpG6$9%-F76#9$@$/F>""!>F>Fep&FgpFYF$F$F$>8.,&FXF;!""F;? (FWF;F;F^q2F;-F76#FboC'>8*-%*addressofG6#-Fjo6$F>,&&FboFJF;FWF;>F^oF;?&8-FboFZ@ &/-Fiq6#-Fjo6$F>,&FarF;FWF;FgqC$>F[oFar>F^oF`o2FdrFgqC&>FgqFdr>F\oFho>&F\oFJFar >F^oFM>FboFco>F\oFho>FRF^r6$-%#opG6#&F>6#;FRF`q-Fhs6#&F>6#;F;-%$maxG6$Fjp,&FRF; F`qF;F$F$F$6" M6R0 I8$7#-%$seqG6$-9!6#8%/FUFH>FMFH@%3/-FJ6$""!F0.%%ProdG/-%%nopsG6#FM """-FJF]o@%-F.6$F0%%listGFM-Fen6#F_oF+F+F+6" M6R0 I;combstruct/draw/tagged_objR6$%%specG%%sizeG6(%#ynG%"yG%$sysG%$typG%$objG%)ato mlistG6#%_pCopyright~(c)~1995~by~Eithne~Murray,~Paul~Zimmermann~and~INRIA.~All~ rights~reserved.G6"C+>8%&9$6#""">8&&F56#""#>8'&F56#""$>8)-%3combstruct/compileG 6$F9F>>8$-&&%1combstruct/CountGFF6#F36#9%@$/FH""!YQCthere~is~no~structure~of~th is~size6">8(-&&%0combstruct/DrawGFFFM6$FO--%5combstruct/draw/RandG6#FHF0@$/F>.% )labelledGC$>FW-%6combstruct/draw/labelG6%7#FWFOFC>FW-%>combstruct/draw/canonic al_labG6#FWFWF0F0F0FU M6R0 I>combstruct/draw/canonical_labR6#%%exprG6"6#%fnCopyright~(c)~1996~Waterloo~Map le~Inc.~All~rights~reserved.GF&-%=combstruct/draw/canonicalizeG6$9$.%)labelledG F&F&F&6" M6R0 I@combstruct/draw/canonical_unlabR6#%%exprG6"6#%fnCopyright~(c)~1996~Waterloo~M aple~Inc.~All~rights~reserved.GF&-%=combstruct/draw/canonicalizeG6$9$.%+unlabel ledGF&F&F&6" M6R0 I4combstruct/finishedR6#%$allG6"6#%^oCopyright~(c)~1995~by~Eithne~Murray~and~IN RIA.~All~rights~reserved.GF&C%@$0%7combstruct/InitializedG%%trueG-%0combstruct/ initGF&@$4-%%typeG6$9$%.iterstructureGYQUargument~must~be~the~result~of~a~call~ to~iterstructs6"&F56#%)finishedGF&F&F&F9 M6R0 I2combstruct/gfeqnsR6%%$SysG%$TypG'%$varG%%nameG67%&assgnG%(numlistG%%varsG%$ls tG%"uG%"FG%&vlistG%%eqnsG%%atomG%$epsG%$attG%(varlistG%*atom_listG%(ep_listG%(m arkepsG%(att_addG%$sysG%$gfnG%$typG%%uvalG%&e_epsG6$%)rememberG%^oCopyright~(c) ~1995~by~Eithne~Murray~and~INRIA.~All~rights~reserved.G6"C,@$0%7combstruct/Init ializedG%%trueG-%0combstruct/initGFC>85-%8combstruct/checkgrammarG6$9$9%>84&FL6 #""">86&FL6#""#>8.<">8/7"@%2""$9#@$5-%%typeG6$&9"6#""%-%%listG6#-Fjo6#F)3-Fco6$ -%$mapG6$R6#%"xGFC6$%)operatorG%&arrowGFC-%'subsopG6$/FV%%NULLGFPFCFCFCFeoFio-F co6$-Fbp6$RFepFCFgpFC-%#opG6$FVFPFCFCFCFeo-Fjo6#<$%'atomicGF)C-@$5/F_oFho4-Fco6 $&Ffo6#""&F)>8$F^q>FjnFeo>80-%%map2G6%FeqFV-%'selectG6$RFepFCFgpFC/-Feq6$FenFP. %%AtomGFCFCFCFS>81-F[s6%FeqFV-F^s6$RFepFCFgpFC/Fbs.%(EpsilonGFCFCFCFS>8%-%'remo veG6$RFepFCFgpFC-Fco6$FdqF)FCFCFCFjn>Fjn-F^s6$RFepFCFgpFCFftFCFCFCFjn>8&-F[s6%F eqFVFjn?&8'FjnFHC'>8(-Feq6$FVFau>8)(%6combstruct/attribute/GFdu>82<#-Feq6#-F[q6 $F]qFau>&836#Fdu6'-%$seqG6$/-Fhu6#8,""!/F]wFir-Fhv6$/-Fhu6#8-F^w/Few-%&minusG6$ FgsF\v-Fhv6$/FcwFV/Few-%*intersectG6$F\vFgs/-Fhu6#FdsF^w/-Fhu6#F^tF^w>&8*Fev7$F duFhu?&FauFatFHC%>87Feu>FewF`v@%4-%)assignedG6#&Fdv6#F^yC%>Fhu(FjuF^y>F\vF]v>Fe y-Fhv6$/-Fhu6#887#F^y/FazFew>Fey6$FeyF\z>Fgn<$-Fhv6$Fcv/FduF]u-Fhv6$&Fdv6#Feu/F auFat>Fjn7#-Fhv6$FhxFjz>FfrF^q>8+-%3combstruct/agfeqnsG6(FSFgnFX9&FjnFfr@$3F]o- Fco6$&Ffo6#F_oF)-%'assignG6$F_\lFfrFf[lFC6%%;combstruct/gfeqns/indcountG%.combs truct/nbGFGFC6" M6R0 I=combstruct/gfeqns/nameinsideR6#%$sysG6&%"xG%&rsideG%"rG%'newsysG6#%^oCopyrigh t~(c)~1995~by~Eithne~Murray~and~INRIA.~All~rights~reserved.G6"C&>8'<"?&8$9$%%tr ueG>-%2combstruct/NameOfG6#-%#opG6$""#F3-F;6$"""F3?&F3F4F5C%>8%F:>8&-%F0-%&unionG6$-FL6$F0<#/F>&FF6#F@&FF6#F=F0F-F-F-6" M6R0 I8&-%#opG6$""!F:@%-F86$FL<$.%+std_constrG .%(usecardGC'@%-F86$FLFX>8*""">Fhn-%%nopsG6#F:>8%F:>8,<"?(8)FinFinFhn%%trueGC$> 8(-FN6$FdoF:@$-F86$FhoFIC%>8$-9!6$FhoFF@%5/-FN6$FPFho%1combstruct/ThetaG/Fgp%/c ombstruct/IntGC%>8'-%2combstruct/NameOfG6#&F`p6#Fin>F_o-%'subsopG6$/FdoF^qF_o>8 +<#/F^qFbqC$>F_o-Ffq6$/FdoFbqF_o>FjqFbo>Fao-%&unionG6%Fao&F`p6#""#FjqO6$F_oFaoY 6$Q7unknown~constructor~%16"FLY6$Q6invalid~production~%1F_sF:F3F3F3F_s M6R0 I4combstruct/gfseriesR6%%$SysG%$TypG'%$varG%%nameG6'%$sysG%$typG%$eqsG%$gfnG%(v arlistG6#%fnCopyright~(c)~Eithne~Murray~and~INRIA.~All~rights~reserved.G6"C)@$0 %7combstruct/InitializedG%%trueG-%0combstruct/initGF2>8'-%8combstruct/checkgram marG6$9$9%>8$&F;6#""">8%&F;6#""#Z%>8&-%2combstruct/gfeqnsG6%FBFG&9"6#;""$!""F2Y F2>8(7#-%#opG6#-Ffn6$FE&FMFD-%;combstruct/gfseries/seriesG6&FMFG9&FYF26#F6F26" M6R0 I;combstruct/gfseries/seriesR6&%(pre_eqsG%$typG%$varG%(varlistG6)%"xG%"yG%#slG% %neqsG%%oeqsG%(slapplyG%$eqsG6#%FCopyright~(c)~Eithne~Murray~and~INRIAG6"C(>8*- %%subsG6$/%$intG%/combstruct/intG9$>8&7#-%$seqG6$/-%$lhsG6#8$""!/FHF6>8'F6>8(FI ?(F3"""FPF30FLFNC&>FNFL>FL7#-FB6$/-FF6#8%-%'seriesG6$-%%evalG6#-F86%/%)infinity G,&%&OrderGFP""#FP/%$SumG%$sumG-%$rhsGFZ9&/FenF?>8)7#-FB6$/-%#opG6$FIFY-%(unapp lyG6$Ffo-Fap6#9'/FenFL>F?7#-FB6$/FE-Fjn6#-F86$F[p-FgoFGFJ-%&tableG6#-%'expandG6 #FLF3F3F36" M6R0 I6series/combstruct/intR6"F$F$F$@%/-%#opG6#&9"6#"""""!&F+6#""#--%(readlibG6#%+s eries/intG6#F+F$F$F$F$ M6R0 I3combstruct/gfsolveR6&%$sysG%$typG%$varG%%tagsG6#%(alleqnsG6#%fnCopyright~(c)~ Eithne~Murray~and~INRIA.~All~rights~reserved.G6"C%@$0%7combstruct/InitializedG% %trueG-%0combstruct/initGF->8$<#-%#opG6#-&%+combstructG6#%'gfeqnsG6#9"-%8(7#-%#opG6#-F96$"""&9$6#F=>8'-%%map2G6%F97$F=""!F?>8 )-%&evalbG6#/-%&minusG6$-FD6%F9FG-%'indetsG6$F?%)functionG<$.%&UnionG.%%ProdG<" @$FI>8*7#-%$seqG6$-8+6#-F96#F6/F_o-FO6$-FO6$FB%1combstruct/atomsG%4combstruct/e psilonsG@%3FI-%%typeG6$-%'selectG6%%$hasG-FD6%F9""#F?Fin-%$setG6#-%'linearG6#Fi nC$>8&7%F?FfnFfn>8$7"C%>F^q-%@combstruct/gfsolve/classifyeqnsG6%F?FB9%-%(readli bG6#%4combstruct/gfseriesGZ%>Faq-%;combstruct/gfseries/seriesG6&F?9&Fhq7#&F6F@F 3YFaq>8%-%9combstruct/gfsolve/solveG6(F^qFfnFaqFhqF6FB@$/Fgr%%FAILG-%)userinfoG 6%Fep%(gfsolveG%:could~not~find~a~solutionGFgrF3F3F3F3 M6R0 I8$-%&unionG6#-%$seqG6$-% 'indetsG6$9$-%)specfuncG6$%)anythingG8%/F?9%-%&evalbG6#50-F86$F0)-%*identicalG6 #9&F><"-%$hasG6$-%$mapG6$%#opGF0FAF-F-F-6" M6R0 I9combstruct/gfsolve/solveR6(%(alleqnsG%$solG%+gfunseriesG%$varG%(varlistG%)non termsG62%%psolG%%varsG%'fFoundG%"rG%%rsolG%"sG%)diffvarsG%)initcondG%(neweqnsG% "xG%&ueqnsG%%eqnsG%)diffeqnsG%&seqnsG%%solZG%"iG6#%fnCopyright~(c)~1996~Waterlo o~Maple~Inc.~All~rights~reserved.G6"C(>8/&9$6#""">80&FC6#""#>81&FC6#""$>8'9%@(0 FA<"C+>FA-%%subsG6$7#-%$seqG6$/-8-6#-%#opG6#9(F[o/F[o9)FA>82-%'selectG6%%%typeG FA/-%*identicalG6#%"ZG%)anythingG@$0FdoFU>FA-FY6$Fdo-%&minusG6$FAFdo>8%-Fep6$<# -F^o6#Fbo<#F]p@%/FAFU>8$<#FdoC%>Fbq<#-%&solveG6$FAFhp@$/FbqFUC$-%)userinfoG6%FJ %(gfsolveG%>solve~did~not~find~a~solutionGO%%FAILG@$F`p>Fbq<#-Fgn6$-%&unionG6$F [oFdo/F[oFbq>Fbq-FY6$7#-Fgn6$/F[oFjnFaoFbq@%/9&7">8(&FbqFDC%>8&%&falseG?&8)Fbq4 F]tC$>Fis-%6$%)operatorG%&arrowGF>- %&evalbG6#0-F^o6$FEFC-F^o6$FJFCF>F>F>F`t&9"6#;FO!"">F]t-F_u6#30FisFcr0FisF^t@$F atC$-F^r6&FJF`r%8,-%@combstruct/gfsolve/cl assifyeqnsG6%-Fjr6$-%%evalG6#-FY6$FisFG-F_w6#-FY6$<#-Fgn6$/-F^o6$7$FE""!83-%(un applyG6$-F^o6$FJF_x-F^o6$7$FEFEF_x/F_xFisFLFbo9'>FQ-9!6%Fhv-Fjr6$-F_w6#-FY6$Fis FRFisFfu0FGFUC,>8*-Fep6$-%$mapG6$F^o-%'indetsG6$FG-%)specfuncG6$F^p%%diffG<#F]o >8+<#-Fgn6$/--F^o6$F^xF[o6#F^x-FY6$/FixF^x&Ffs6#F[o/F[oFgy>Fbq-%*traperrorG6#-F _w6#-%'dsolveG6$-Fjr6$FGFfzFgy@$/Fbq%*lasterrorGC$-F^r6&FJF`r%5dsolve~gave~an~e rrorGFbqOFcr@$/Fbq%%NULLGC$-F^r6%FJF`r%?dsolve~did~not~find~a~solutionGOFcr>Fbq -%;combstruct/gfsolve/dfiddleG6%<#FbqFfsFix>Fis-Fet6$FbqFfu@$5/FisFcr/FisF^tC$- F^r6&FJF`r%?dsolve's~answer~did~not~verifyGFbqOFcr>Fhv-Fjv6%-F_w6#-FY6$-F[z6$RF jtF>F[uF>/-F^o6$F^xFbu-Fax6$Fdu-F^o6$FEFbuF>F>F>FisFLFboFix>FQ-F\y6%Fhv-Fjr6$-F _w6#-FY6$FbqFRFbqFfu0FLFUC&>8.-Ffo6%R6$F5%$ntsGF>F[uF>-%$hasG6$-F^z6$Fdu%)funct ionGFRF>F>F>FLFbo@$0F^`lFL-%&printG6#%Gack!~not~the~way~I~thought~this~workedG- F^r6&FJF`r%:subs~equations~unresolvedGFLOFcrFQF>F>F>6" M6R0 I8,-%'indetsG6$F=-%)specfuncG6$-%(polynomG6$%)anythingG%#_ZGF>>8*-%'s electG6$R6#F0F76$%)operatorG%&arrowGF7-%&evalbG6#2""#-%'degreeG6$-%#opG6$"""F=F MF7F7F7FB@%/FOFB>8+F=C&>Fao-%%subsG6$7#-%$seqG6$/8)-%2combstruct/ROOTOFG6#-F[o6 #F\p/F\pFOF=>Fao-%*traperrorG6#-%*allvaluesG6#Fao@$/Fao%*lasterrorGC$-%)userinf oG6&Ffn%(gfsolveG%8allvalues~gave~an~errorGFaoO%%FAILG>Fao-F[o6#-Feo6$/F^pF><#F ao>FaoF=>8(7#-Fio6$/F\p-%(unapplyG6$&9%6#-F\p6#-F[o6#9'F[s/F\p9(>%)TestzeroG%%+NormalizerGRFTF76#%fnCopyright~(c)~1996~Waterloo~ Maple~Inc.~All~rights~reserved.GF7@%/-Fas6#-%&evalcG6#F=%%trueG""!F=F7F7F7-%'as sumeG6$2F`t8-2Fet#F]o"(+++"?&8%F[rF_tC%>8$%&falseG?&8'FjtF_tC&@$-F;6$-F[o6$FfnF `uF_s>F`u/-F[o6$F]oF`u-%%evalG6#-Feo6$F^rFeu>8&-Fep6#-%'normalG6#-%'expandG6#-% $mapG6$F]t-%'seriesG6$,&-F\v6#-Feo6'/%)infinityG,&%&OrderGF]oFfnF]o/%$SumG%$sum G/%$intG%$IntG/9&FetFeuF]o-Feo6$F_x&Fgr6#Fiu!""Fet@$/FavF\qC$-F_q6&FfnFaq%5seri es~gave~an~errorGFavOFdq@$330FavF`t4-%%typeG6$Fav-%"OG6#FL0-%%nopsG6#FavFfnC$>F ]uF_t[@$4F]uC$@$/%-_EnvExplicitGF_t>Fjt-F\v6#-Feo6$FiwFjtOFjtF^uF7F7F76" M6R0 I;combstruct/gfsolve/dfiddleR6%%(dresultG%+gfunseriesG%$varG6)%&constG%'badeqsG %#eqG%"sG%$solG%"rG%"xG6#%fnCopyright~(c)~1996~Waterloo~Maple~Inc.~All~rights~r eserved.G6"C&>8$-%'selectG6$R6#F/F26$%)operatorG%&arrowGF2/-%*substringG6$9$;"" """#%#_CGF2F2F2-%'indetsG6$FB.%'symbolG>8%-F76%%$hasGFBF5@%0F5<"C%>8'<#-%$seqG6 $-%'seriesG6$,&-%#opG6$FE8&FD&9%6#-Fjn6$FDF\o!""9&/F\oFM>8(<#-FY6$/-Fjn6$FD8*-% (convertG6$-Fjn6$FEF]p%(polynomG/F]p-%&solveG6$FVF5>8)-%%subsG6$FfoFB>FipFBFipF 2F2F26" M6R0 I@combstruct/gfsolve/classifyeqnsR6%%(alleqnsG%)nontermsG%$varG6'%&deqnsG%&seqn sG%&ieqnsG%%eqnsG%)diffeqnsG6#%fnCopyright~(c)~1996~Waterloo~Maple~Inc.~All~rig hts~reserved.G6"C)>8%-%'selectG6&%8'-%&minu sG6$F8F3>8$-F56%%(hastypeGF<-%)specfuncG6$%)anythingG%%diffG>8&-F56%FDF<-FF6$FH %$intG>F<-F>6$-F>6$F8(-%&unionG6$FA-%$mapG6%FIFKF:7%F8$9$-F66$F87$F;F;C$@$3333-F66$7$F>9%7$%) functionGFM/-%#opG6$""!F>-FP6$FRFK/-%%nopsG6#F>"""/-FW6#FKFY-F"6$-FPFX-FPFfnO%% trueG>F=,&F>FYFK!""YQ1Invalid~argumentF2@-/F=FROF\o3-F66$F=%"^G-F66$-FP6$""#F=% (numericG@%2FRF[pO-F"6#-FP6$FYF=O%&falseG-F66$F=%"*GC$?&8%F=F\oC$>8+-F"6#F]q@&/ F`qF\oOF\o/F`q%%FAILGOFgqOFgp-F66$F=FM@*/-FP6$FRF=%$expGOFgp/F^r%#lnGO-F"6#,&Fd pFYF_oFY/F^r%$absGO-F"6#-FP6#F=/F^r%"OGOFgp-%$hasG6$F=<$%"LG%"QGO-F"6#-%%evalG6 #-%%subsG6%/FfsR6#%"xGF2F2F2-Fcr6#*$,&FYFYF>F_oF_oF2F2F2/FgsRFctF2F2F2FgtF2F2F2 F=C$>8'-%&numerGF^s@$2-%'lengthG6#F]u-FcuF^sO-F"Fdu@$-F66$F=-%(ratpolyG6#%)rati onalGO-%&evalbG6#Fco>8*-%(convertG6$F=.%'RootOfG@$-F66$Fdv%'algnumGC$>F]u-%*tra perrorG6#-%&evalaG6#-%'NormalG6#Fdv@&0F]u%*lasterrorGO-Fav6#/F]uFR-F66$7#F]u7$- %*identicalG6#%Mreducible~RootOf~detected.|*Substitutions~areG%$setG@$/-FW6#-%' indetsG6$FdvFhvFYC%>F]u-FP6$F]pFbx?&F]qF]uF\o@$/-F"6#-F_t6$F]qFdvF\oOF\oOFgp>8& -%&minusG6$-F^y6$F=%%nameG<#%*constantsG@%0F^z<"C$>%'DigitsG,&F\[lFY""&FY?(8(FY FY""$F\oC&@'/F`[lFY>F]u-Faw6#-F_t6$-%$seqG6$/-FP6$F]qF^z-%&evalfG6#)#"%*y""%#\" *$F]qF_o/F]q;FY-FW6#F^zF=/F`[lF]p>F]u-Faw6#-F_t6$-F[\l6$/F^\l-Fa\l6#)#Ff\lFe\lF ]qFh\lF=>F]u-Faw6#-F_t6$-F[\l6$/F^\l,$F`\lF_oFh\lF=@$4-F66$-Faw6#-Fa\lFduF^p\>F \[l,&F\[lFY!"&FY@%-%'memberG6$-Faw6#--%(readlibG6#%&evalrG6#-%'SignumGFdu<$F_oF YOFgp[C$>F]u-%&evalcGF^s@(/F]uF=@$-F__l6$-Faw6#-Fd_l6#-Fj_lF^sF[`lOFgp0-F"6#-%& coeffG6%F]u^#FYFRF\oOFgp0-F"6#-Faal6%F]uFcalFYF\oOFgp>F]u-%'testeqGF^s@$0F]uFgq OF]u@&3/F^zFiz4-%(hastypeG6$F=%(nonrealGC$-%)userinfoG6&Fa[l.%+equivalentG%9War ning:~assumption~madeGFcoOF\o-FcsFgvC$>F^z-F^yFgv@$/Fj\lFYO-F"6$-F_t6$/-FPF[]l8 )F^u-F_t6$/%#_ZGF^dl-FP6$FYF]dlOFgqF2F2F26" M6R0 I0combstruct/initR6"F$6#%_pCopyright~(c)~1995~by~Eithne~Murray,~Paul~Zimmermann ~and~INRIA.~All~rights~reserved.GF$C$@$0%7combstruct/InitializedG%%trueGC%@$4-% )assignedG6#%3combstruct/usegfunG>F2%&falseG@$4-F06#%2combstruct/methodG>F9.%.b oustrophedonG>F*F+%%NULLGF$6&F*%3combstruct/versionGF2F9F$6" M6R0 I6combstruct/parse_argsR6$%*StructureG%%SizeG6&%*structypeG%'structG%"sG%$gnfG6 $%)rememberG%_pCopyright~(c)~1995~by~Eithne~Murray,~Paul~Zimmermann~and~INRIA.~ All~rights~reserved.G6"C&@'/9#""#@%4-%%typeG6$9%/-%*identicalG6#.%%sizeG<%%*non negintG-F=6#.%)allsizesG-F=6#.%(defaultGY6$Q>invalid~size~specification~%16"F:> 8&-%#opG6$F4F:/F3""">FPFIYQ8%9$@%-F86$FZ%%listGC )@$/-%%nopsG6#FZF4>FZ7$-FRF_o.%+unlabelledG@$0F]o""$YQSwrong~number~of~argument s~in~grammar~specificationFN>8'-%8combstruct/checkgrammarG6$&FZ6#F4&FZ6#Fgo@$4- %'memberG6$&FZ6#FU-%'indetsG6$F_p%%nameGY6$QYnon-terminal~%1~does~not~appear~in ~grammar~specificationFNFhp>8$.%(grammarG>FZ7$Fhp-FR6#F[p@$4-F86$FPFBYQ8invalid ~or~missing~sizeFN>Fbq.%(specialG7%FbqFZFPF/F/F/FN M6R0 I-type/usecardR6#%"xG6"6#%_pCopyright~(c)~1995~by~Eithne~Murray,~Paul~Zimmerman n~and~INRIA.~All~rights~reserved.GF&-%'memberG6$9$<&.%$SetG.%&CycleG.%)Sequence G.%)PowerSetGF&F&F&6" M6R0 I0type/std_constrR6#%"xG6#%"rG6#%_pCopyright~(c)~1995~by~Eithne~Murray,~Paul~Zi mmermann~and~INRIA.~All~rights~reserved.G6"C$@%-%%typeG6$9$%(indexedG>8$-%#opG6 $""!F0>F3F0-%'memberG6$F3<*.%)PowerSetG.%&UnionG.%%ProdG.%&SubstG.%1combstruct/ ThetaG.%/combstruct/IntG.%1combstruct/DeltaG.%0combstruct/PSetGF*F*F*6" M6R0 I+type/deltaR6#%"xG6"6#%_pCopyright~(c)~1995~by~Eithne~Murray,~Paul~Zimmermann~ and~INRIA.~All~rights~reserved.GF&-%%typeG6$9$&%1combstruct/DeltaG6#<'%(integer G-%$SetG6#F1-%*identicalG6#.F3-%&CycleGF4-F66#.F:F&F&F&6" M6R0 I-type/productR6#%"xG6"6#%_pCopyright~(c)~1995~by~Eithne~Murray,~Paul~Zimmerman n~and~INRIA.~All~rights~reserved.GF&-%&evalbG6#/9$.%%ProdGF&F&F&6" M6R0 I1type/nonterminalR6#%"xG6"6#%_pCopyright~(c)~1995~by~Eithne~Murray,~Paul~Zimme rmann~and~INRIA.~All~rights~reserved.GF&-%&evalbG6#3-%%typeG6$9$%%nameG4-%'memb erG6$F0<$.%%AtomG.%(EpsilonGF&F&F&6" M6R0 I*type/atomR6#%"xG6"6#%_pCopyright~(c)~1995~by~Eithne~Murray,~Paul~Zimmermann~a nd~INRIA.~All~rights~reserved.GF&-%'memberG6$9$%1combstruct/atomsGF&F&F&6" M6R0 I-type/epsilonR6#%"xG6"6#%_pCopyright~(c)~1995~by~Eithne~Murray,~Paul~Zimmerman n~and~INRIA.~All~rights~reserved.GF&5-%'memberG6$9$%4combstruct/epsilonsG-%%typ eG6$F-.%)uniontagGF&F&F&6" M6R0 I+type/substR6#%"xG6"6#%_pCopyright~(c)~1995~by~Eithne~Murray,~Paul~Zimmermann~ and~INRIA.~All~rights~reserved.GF&-%&evalbG6#/9$.%&SubstGF&F&F&6" M6R0 I2combstruct/NameOfR6#%%exprG6$%"rG%%remtG6#%_pCopyright~(c)~1995~by~Eithne~Mur ray,~Paul~Zimmermann~and~INRIA.~All~rights~reserved.G6"C$>8%-%#opG6$8$-%%evalG6 #9!@%30F.%%NULLG-%)assignedG6#&F.6#9$F>C%>F2-%3combstruct/newnameGF+-%)userinfo G6(""#%+combstructG%5Introducing~new~typeGF2%$forGF@>-F6F?F2F+F+F+6" M6R0 I3combstruct/newnameR6"F$6#%_pCopyright~(c)~1995~by~Eithne~Murray,~Paul~Zimmerm ann~and~INRIA.~All~rights~reserved.GF$C$>%.combstruct/nbG,&F)"""F+F+(%-combstru ct/TGF)F$6#F)F$6" M6R0 I.combstruct/nb!""6" M6R0 I3type/iterstructureR6#%"tG6"6#%_pCopyright~(c)~1995~by~Eithne~Murray,~Paul~Zim mermann~and~INRIA.~All~rights~reserved.GF&3-%%typeG6$&9$6#.%*nextvalueG%*proced ureG-F+6$&F.6#.%)finishedG%(booleanGF&F&F&6" M6R0 I.type/uniontagR6#%"tG6"6#%_pCopyright~(c)~1995~by~Eithne~Murray,~Paul~Zimmerma nn~and~INRIA.~All~rights~reserved.GF&3-%%typeG6$9$.%'symbolG-%&evalbG6#/-%*subs tringG6$F-;"""-%'lengthG6#%4combstruct/uniontagGF8(-%6combstruct/parse _argsG6#9"@$-%%typeG6$&F76#"""-%*identicalG6#.%(grammarGYQ7grammars~not~support ed6">8)&F76#""#>8*&F76#""$@'3-F>6$FL.%)functionG-F>6$-%#opG6$""!FL.%%nameGC$>8% Fgn>8&6$FQ-Fhn6#FL-F>6$FLF[oC$>F_oFL>FaoFQY6$Q:invalid~structure~name~%1FJFL>8' (%8combstruct/iterstructs/GF_o@$4-F>6$F^p.%*procedureGC$>8$-%*traperrorG6#-%(re adlibG6#F^p@$/Fip%*lasterrorGY6$QAnot~implemented~for~structure~%1FJFL-F^p6#Fao F$F$F$FJ M6R0 I6combstruct/nextstructR6#%$allG6"6#%^oCopyright~(c)~1995~by~Eithne~Murray~and~ INRIA.~All~rights~reserved.GF&C&@$0%7combstruct/InitializedG%%trueG-%0combstruc t/initGF&@$4-%%typeG6$9$%.iterstructureGYQUargument~must~be~the~result~of~a~cal l~to~iterstructs6"@$&F56#.%)finishedGYQRattempt~to~call~nextstruct~on~a~finishe d~iteratorF9-&F56#.%*nextvalueGF&F&F&F&F9 M6R0 I8combstruct/standardformR6$%$SysG%$typG6$%"xG%$sysG6$%)rememberG%_pCopyright~( c)~1995~by~Eithne~Murray,~Paul~Zimmermann~and~INRIA.~All~rights~reserved.G6"C(> 8%-%&unionG6#-%$seqG6$-%Ccombstruct/standardform/NameInsideG6$8$%%trueG/F:9$>%8 combstruct/user_sys_NF1GF0-%'forgetG6$%2combstruct/NameOfG/%-subfunctionsG%&fal seG>F0-%Hcombstruct/standardform/GreeneTransformG6$F09%>F0-F26#-F56$-%@combstru ct/standardform/binProdG6#F:/F:F0F0F-6#F?F-6" M6R0 I@combstruct/standardform/binProdR6#%#eqG6"6#%fnCopyright~(c)~1996~Waterloo~Map le~Inc.~All~rights~reserved.GF&@%0-%'indetsG6$9$-%)specfuncG6$%)anythingG%%Prod G<"-%Ccombstruct/standardform/NameInsideG6$-%%evalG6#-%%subsG6$/F3%=combstruct/ standardform/ProdGF.%&falseG<#F.F&F&F&6" M6R0 I=combstruct/standardform/ProdR6"F$6#%_pCopyright~(c)~1995~by~Eithne~Murray,~Pa ul~Zimmermann~and~INRIA.~All~rights~reserved.GF$@%/9#""#-%%ProdG6#9"-F,6$&F.6#" ""-9!6#&F.6#;F*F)F$F$F$6" M6R0 IHcombstruct/standardform/GreeneTransformR6$%$SysG%$typG6&%$sysG%"xG%$newG%#eqG 6#%_pCopyright~(c)~1995~by~Eithne~Murray,~Paul~Zimmermann~and~INRIA.~All~rights ~reserved.G6"C&>8$9$@$-%$hasG6$F1.%)PowerSetG>F1-%Acombstruct/standardform/Powe rSetG6$F19%?(F."""F?F.0-%'indetsG6$F1-%)specfuncG6$%)anythingG<%.%$SetG.%&Cycle G.%)SequenceG<"?&8'F1%%trueG@$-%%typeG6$-%$rhsG6#FQFDC%>F1-%&minusG6$F1<#FQ>8&- &%?combstruct/standardform/GreeneG6$-%#opG6$""!FWF=6$-%$lhsGFY-Fao6#FW>F1-%&uni onG6$F1-%$seqG6$-%Ccombstruct/standardform/NameInsideG6$8%%&falseG/FcpF[o-%%eva lG6#-%%subsG6)/%&UnionGR6#F)F.F.F.@%/9#F?F2.-F]q6#9"F.F.F./%%ProdGRF_qF.F.F.@%F aqF2.-FhqFeqF.F.F./-FJ6#F?FI/-FLF_rFK/&%1combstruct/DeltaGF_rRF_qF.6$%)operator G%&arrowGF.F2F.F.F./%1combstruct/ThetaGRF_qF.F.F.@%-FU6$F2.%%atomGF2.-Fjr6#F2F. F.F.F1F.F.F.6" M6R0 I?combstruct/standardform/Greene=6"%&falseGE\[l'6$%$SetG%+unlabelledGR6%%"AG%"B G%(cardresG6)%"jG%%initG%"kG%"iG%"lG%"TG%'newrhsG6#%_pCopyright~(c)~1995~by~Eit hne~Murray,~Paul~Zimmermann~and~INRIA.~All~rights~reserved.GF$@%/9#""#C$>8)-%2c ombstruct/NameOfG6#-F(6$9%1""".%%cardG<$/9$-%&UnionG6$.%(EpsilonGF?/F?FC@'-%*ty pematchG6$9&1'8&%(integerG-%*identicalG6#FHC&>F?F@>8%/F?-%/combstruct/IntG6#-FN 6$-&%1combstruct/DeltaG6#F(6#-%1combstruct/ThetaG6#FE-%%ProdG6$FcoF?@'/FZ""!>F\ o6$FKF\o/FZFG@$0FLF?>F\o6$/FLF?F\oC%>8(-%&arrayG6#;F`pFZ>&F[q6#F`pF??(8'FGFGFZ% %trueGC$@%/FdqFZ>&F[q6#FdqFL>Fjq-%3combstruct/newnameGF$>F\o6$F\o/Fjq-F_o6#-FN6 %-%$seqG6$-F\p6$-&Feo6#8$Fgo&F[q6#,&FdqFGF^s!""/F^s;FG,&FdqFGFbsFG-&Feo6#-F(F[r Fgo-F\p6$FfsFaq<#F\o-FU6$FW/FfnFY@'F_p<#/FLFPFcp<#/FLFEC(>F[qF\q>FaqFP>&F[q6#FG FE>F\o%%NULLG?(FdqFFjqFL>FjqF]r>F\o6$F\o/Fjq-F_o6#-FN6$Ffr-&Feo F[rFgoF\tC$-FU6$FW1FfnFY@%Fcp<#/FL-FN6$FPFEC&>F[q-F]q6#;FGFZ>F\oF\u?(FdqFGFGFZF eqC$>8*-F_o6#-FN6$-Fgr6$-FN6$F[sFirFcsFiu@%Fhq>F\o6$F\o/FL-FN6$F]wFPC$>FjqF]r>F \o6$F\o/FjqF]wF\tF$F$F$6$F(%)labelledGRF+6*%"pG%"xGF1%#A0G%%totoGF2F5F0F7F$@%F: C$>F]wF@<$/FL-FN6$FPF]w/F]wFC@'-FU6$FW1'F?FenFfnC$>F]wF@@'/F?F`p<$/F]w-FN6$FE-F _o6#-F\p6$FhoF]wF]y/F?FG@%-%'memberG6$FL<#-Fgr6$-%$lhsG6#F\o/F\o%8combstruct/us er_sys_NF1G<$/FLF]wF[z<#/FL-FN6$FE-F_o6#-F\p6$FhoFLC(>FdqF]w>FZF[z>F^s-F_o6#-F\ p6$FhoF\o>F[qF\o?(F$FGFG,&F?FGFbsFGFeq>F[q-%%subsG6$/F\oF^sF[q<$FZ/FL-Ff\l6$/F\ oFdqF[q-FU6$FW/FfnFeyC'@%Fiy>FdqFP>FdqF\u>F^sF]\l>F[qF\o?(F$FGFGF?Feq>F[qFe\l<# Fj\lC$-FU6$FW1FfnFey<#/FL-FN6$FP-Fgr6$-F(6$FE/FH8+/Fg^l;FGF?F$F$F$6$%&CycleGF)R F+6&F2F0%"rGFfxF7F$C$@)F:>FZ-&Feo6#F[_l6#-FN6$Fho-F\p6$Fho-%)SequenceGFD-FU6$FW 1'F^sFenFfn>FZ-FN6$-Fgr6$-%"$G6$-&FeoF\[l6#-F\p6$Fho-F[`l6$FE1-%%ceilG6#,&*&F^s FGF\oFbsFGFbsFGFH-&%*numtheoryG6#%$phiGF\[l/F\o;FG,&F^sFGFbsFG-&Feo6#-F[_lF]sFe _l-FU6$FW/FfnF_`l>FZ-FN6#-Fgr6$-Ff`l6$-Fi`l6#-F\p6$Fho-Ff`l6$FEFcalFeal/F\o-&Fg al6#%)divisorsGF]sC%-FU6$FW1FfnF_`l?(F\oFGFGF^sFeq@%/,&-%%iquoG6$F^sF\oFGFbsFGF `p>&FZF\[l-Ff`l6$-Fi`lFgoFeal>Fbdl-Fgr6$-%#opGF[r/Fdq7#-Ff`l6$7#-Fi`l6#-FN6$Fho -F\p6$Fho-%Dcombstruct/standardform/PosSequenceG6$FEF]dlFeal>FZ-FN6#-Fjdl6#-%(c onvertG6$FZ%%listG<#/FL-F_o6#-Fjdl6#-%%evalG6#-Ff\l6$/F[`l%Ccombstruct/standard form/SequenceNFG<#FZF$F$F$6$F[`lFbxRF+6*F2FexFfxF1FgxFhxF5F3F7F$@%F:C$>F]w-FA6# Fj_l<$F]y/F]wFj_l@'F\`lC$>F]wFfgl@'/F^sF`p<$/F]w-FN6$FE-F\p6$FEF]wF]y/F^sFG@%-F ez6$FL<#-Fgr6$-F[[l6#FZ/FZF^[l<$F`hlF`[l<#/FL-FN6$FE-F\p6$FEFLC(>FdqF`hl>F[qF]w >F\o-F\p6$FEFZ>F?FZ?(F$FGFGF\blFeq>F?-Ff\l6$/FZF\oF?<$Fdq/FL-Ff\l6$/FZF[qF?Fabl C(@%F^hl>F[qFP>F[qF\u>FdqF\u>F\oFjil>F?FZ?(F$FGFGF^sFeq>F?F_jlFbjlC$Fgcl<#/FL-F N6%FPFE-Fgr6$-F\p6$FE-Ff`l6$FEFg^l/Fg^lF[blF$6#F^[lF$6$F[`lF)Fagl6$F[_lFbxRF+6# F2F7F$@)F:<#/FL-FN6$FE-F_o6#Fh_lF\`l@%Fehl-9!6$FLFE<#/FL-F_o6#-F\p6$Fho-F[`l6$F E1F\blFHFabl@%FehlFct<#/FL-F_o6#-F\p6$Fho-F[`l6$FE/FHF\blC$Fgcl<#/FL-FN6$FE-F_o 6#-F\p6$Fho-Fgel6$FEF\blF$F$F$6" M6R0 IDcombstruct/standardform/PosSequenceR6$%"AG'%"kG%'posintG6#%"iG6#%fnCopyright~ (c)~1996~Waterloo~Maple~Inc.~All~rights~reserved.G6"-%&UnionG6$9$-%$seqG6$-%%Pr odG6$F1-%"$G6$F18$/F;;""",&9%F>!""F>F-F-F-6" M6R0 ICcombstruct/standardform/NameInsideR6$%#eqG%%fTagG6'%"rG%#rsG%"xG%"iG%'newepsG 6#%_pCopyright~(c)~1995~by~Eithne~Murray,~Paul~Zimmermann~and~INRIA.~All~rights ~reserved.G6"@%-%%typeG6$-%#opG6$""#9$<%%%nameG-%*identicalG6#.%(EpsilonG-F<6#. %%AtomG<#F8C&>8%F4@%39%/-F56$""!FG.%&UnionGC$>8(<#-%$seqG6$/-%Acombstruct/stand ardform/UnionTagGF/F>/8';"""-%%nopsG6#FG>FG-FP6#-FV6$-%%ProdG6$-F56$7$FfnFhnFS- F56$FfnFGFen>FS<">8$7#-FV6$-%@combstruct/standardform/InnerNIG6$8&FJ/FcpFG-%&un ionG6$-Ffp6$<#/-F56$FhnF8-FL6#-FV6$&Fcp6#Fhn/FcpF\p<#-FV6$-F56#&Fcp6#F7FdqFSF/6 #%Ccombstruct/standardform/UnionTagCtGF/6" M6R0 I@combstruct/standardform/InnerNIR6$%%exprG%%fTagG6*%"rG%"xG%&newrsG%)newprodsG %&newntG%#rsG%"iG%'newepsG6#%_pCopyright~(c)~1995~by~Eithne~Murray,~Paul~Zimmer mann~and~INRIA.~All~rights~reserved.G6"C$>8)9$@%-%%typeG6$F5<&%)relationG%%name G-%*identicalG6#.%(EpsilonG-F?6#.%%AtomG7$F5<"C(@%39%/-%#opG6$""!F5.%&UnionGC$> 8+<#-%$seqG6$/-%Acombstruct/standardform/UnionTagGF2FA/8*;"""-%%nopsG6#F5>F5-FS 6#-FY6$-%%ProdG6$-FO6$7$FinF[oFV-FO6$FinF5Fhn>FVFH>8$7#-FY6$-9!6$8%FL/FepF5>8&- FN6#-FY6$&Fep6#F[o/FepF^p>8'-%&unionG6#-FY6$&Fep6#""#F_q>8(-%2combstruct/NameOf G6#Fhp7$F[r-Fcq6$-Fcq6$<#/F[rFhpFVFaqF26#%Ccombstruct/standardform/UnionTagCtGF 26" M6R0 IAcombstruct/standardform/PowerSetR6$%$SysG%$typG6%%#eqG%"TG%$sysG6#%fnCopyrigh t~(c)~1996~Waterloo~Maple~Inc.~All~rights~reserved.G6"C%>8&9$@%/9%.%)labelledG> F0-%%evalG6#-%%subsG6$/.%)PowerSetG.%$SetGF0?&8$F0%%trueG@$-%%typeG6$-%$rhsG6#F D-%)specfuncG6$%)anythingG<#F?C%>8%-%2combstruct/NameOfG6#-FB6#-%#opG6#FJ>F0-%& minusG6$F0<#FD>F0-%&unionG6$F0<$/FTFX/-%$lhsGFL-%0combstruct/PSetG6$FTFZF0F-F-F -6" M6R0 IAcombstruct/standardform/UnionTagR6"6$%"rG%"iG6#%_pCopyright~(c)~1995~by~Eithn e~Murray,~Paul~Zimmermann~and~INRIA.~All~rights~reserved.GF$C&>8$<#-%'unamesGF$ ?(8%,&%Ccombstruct/standardform/UnionTagCtG"""F4F4F4F$5-%)assignedG6#(%4combstr uct/uniontagGF1-%'memberG6$F9F,F$>F3F1F9F$6#F3F$6" M6R0 ICcombstruct/standardform/UnionTagCt""!6" M6R0 I3combstruct/version$"#T!""6" M6R0 I+combstruct=6"%&falseGE\[l-%+nextstructG%6combstruct/nextstructG%+allstructsG% 6combstruct/allstructsG%(agfeqnsG%3combstruct/agfeqnsG%)gfseriesG%4combstruct/g fseriesG%*agfseriesG%5combstruct/agfseriesG%'gfeqnsG%2combstruct/gfeqnsG%%drawG %0combstruct/drawG%/agfmomentsolveG%:combstruct/agfmomentsolveG%&countG%1combst ruct/countG%,iterstructsG%7combstruct/iterstructsG%(gfsolveG%3combstruct/gfsolv eG%)finishedG%4combstruct/finishedG%*protectedG M6R0 I-encyclopedia=6"%&falseGE\[l'%-computeentryGR6''%%nameG%'stringG'%.specificati onG7%F+%$setG<&-%*identicalG6#%+unlabelledG-F36#%)labelledG-F36#%*unlabeledG-F3 6#%(labeledG'%,DescriptionGF,'%+referencesGF,'%'optindG%(integerG6/%,CountingSe qG%#GFG%"iG%/specification1G%"SG%+RecurrenceG%$DEqG%+ClosedFormG%'FormatG%'Shif t1G%+AsymptoticG%"nG%+shiftedrecG6#%fnCopyright~Stephanie~Petit,~Inria~Rocquenc ourt,~France,~1998GF$C+>8'-%%subsG6$/&9%6#"""8(Fhn>8$7#-%$seqG6$-&%+combstructG 6#%&countG6$Fhn/%%sizeG8&/Fjo;""!"#?>8)%%FAILG>8+Fap>8.Fap>8%-&Fdo6#%(gfsolveG6 %&FX6#""#&FX6#""$%#_xG-%'forgetG6#%2combstruct/gfeqnsG@$0FgpFapC%>Fgp-FZ6$Fgp-F [o6#FcqZ%C&>8*-&%%gfunG6#%0holexprtodiffeqG6$Fgp-%"yGF_r>F`p-&Ffr6#%,diffeqtore cG6%FcrFjr-%#_fG6#%#_nG@$/&FhnFaqF8>F`p-&Ffr6#%)invborelG6$F`pFbs@%-%%typeG6$Fg p%(ratpolyGC$>Fcp-%)simplifyG6%-&Ffr6#%/ratpolytocoeffG6%FgpFcqFes.%&powerG.%(r adicalG@$-%'memberG6$Fhs<$.F8.F>>Fcp*&FcpFjn-%*factorialGFdsFjnC(>8,-&Ffr6#%*fo rmatrecG6&7$F`pFbsFcsFes.%%initG>8-,(-%%nopsG6#FgvFjn-F\w6#F_v!""F_qFjn?(FjoF_q FjnF$/&F_v6#FjoF]p>Fiv,&FivFjnFjnFjn@%2F]pFiv>80-_Ffr%(makerecG6&-FZ6$/Fes,&Fes FjnFivFjnF_vFcsFes-%'removeG6%Fat<#-F`o6$/-Fcs6#,&-%#opG6$7$FjnFjnFjoFjnFivF`w- F^y6$F_qFjo/FjoFgv/-Fcs6#%'negintG%)anythingG>FjwF`p>Fcp-%'rsolveG6$FjwFbs@%-Fa t6$-F^y6$FjnFcpF0>FcpFapC$>Fcp-FZ6$/Fes,&FesFjnFivF`wFcp>Fcp-Fgt6&FcpFesF^uF`uF $F$Z%C$>Fep-%%evalG6$-%+equivalentG6%-FZ6$/Fcq%"xGFgpFi[l8/7$/%"OGF]p/Fj[lFes@$ -%$hasG6$Fep%-_saddlepointG>FepFapF$F$-%(EnclineG6,9$FXF]oFgpF`pFcpFep9&9'-%#if G6%/9#""&9(%%NULLGF$F$F$%*seqsearchGR6#%%ListG6*FI%#nlG%&List1G%&rListG%'result G%"TG%"rG%$outGFTF$C)@%2F^p-F\w6#Fh\lC$>Fjo&Fh\l6#;FjnF^p>FX7#-F`o6$&Fh\l6#,&F^ pFjnF]oF`w/F]o;F]p"#>C%>FjoFh\l>FgpFb^l>FX7#-F`o6$&Fh\l6#,&FgpFjnF]oF`w/F]o;F]p ,&FgpFjnF`wFjn>F`p&%1encyclopedia/indG6#""%>F[o<#-F^y6#&F`p6#&FXFin?(F]oF_qFjn- F\w6#FX%%trueG>F[o-%*intersectG6$F[o<#-F^y6#&F`p6#&FX6#F]o?&FcrF[oF_al@%-%6ency clopedia/ContainsG6$-F^y6$FbqFcrFjo>&Fcp6#FcrFcr>FcblFb]l>F[o<#-F`o6$Fcbl/FcrF[ o@%/F[o<"Fap7#-F^y6#F[oF$F$F$%)gfsearchGR6#FH6&FI%$GF1G%$varGF\^lFTF$C%>Fjo-%&m inusG6$-%'indetsG6$Fh\lF+<#%*constantsG>FX&Fb`lFin@%/-F\wFdwFjnC$>Fgp-%'normalG 6#-FZ6$/-F^yFdwFcqFh\l@%-Fat6$&FX6#Fgp%%listGFbelFap-%&ERRORG6$QInot~a~closed~f orm~with~a~single~variableF$Fh\lF$F$F$%)nbsearchGR6#%#nbGF$F$F$@%4-%)assignedG6 #&%7encyclopedia/maintableGFc^l-Ffel6$Fh\lQ7Not~a~valid~ECS~numberF$7#FbflF$F$F $%/keywordssearchGR6#%%ExprG6'F]^l%%wordG%#WLGFf]lF\^lFTF$C%>Fjo-%6encyclopedia /wordlistGFc^l>F[o&Fb`lFaq@%-F`fl6#&F[o6#-F^y6$FjnFjoC%>FX<#-F^yFggl?&Fgp-Fjcl6 $Fjo<#FjglF_al>FX-Fbal6$FX<#-F^y6#&F[oFcel@%/FXF]clFap7#-F^yF^alFapF$F$F$%)cfse archGR6#FN6&FI%,ClosedForm1GFfclF\^lFTF$C%>FjoFicl>FX&Fb`lF^q@%FddlC%>FjoF^el>F gp&FX6#-Fgt6&-FZ6$/FjoFesFh\lF^uF`u.%&GAMMAG@%-Fat6$FgpFdelFgpFap-Ffel6$FhelFd` lF$F$F$F$ M6R0 I3encyclopedia/indGF=6"%&falseGE\[ldcn*&,(*&%#_xG"""-%'RootOfG6#,(%#_ZG!"'F*""' *&F*F+F0""$F+""#F5*&F*F5F,F+F4!")F+F+,&F7F+*$F*F4""*!""7#-%(EnclineG6,QILabelle d~3-constrained~functional~graphsF$7%%"SG<$/FB-%$SetG6#-%&CycleG6#-%%ProdG6$%"Z G-FF6$%"gG/%%cardGF5/FQ-%&UnionG6$FN-FL6$FN-FF6$FQ/FSF4%)labelledG77F+""!FinF4F inFin"$+$FinFin"'?69FinFin"*+?&H=FinFin"-+S]3]]FinFin"1+?jLlWZDFinFin,$*$,&!"#F +F)F+F;Fbo<)/-%#_fG6#FinF+/-Ffo6#F4F4/-Ffo6#F+Fin/-Ffo6#F5Fin/-Ffo6#""%Fin/-Ffo 6#""&Fin,&*&,,*$%#_nGF5!$<"F]q!$3"*$F]qF4!#a!#OF+*$F]qFdp!"*F+-Ffo6#F]qF+F+*&,& F]q"")"#7F+F+-Ffo6#,&F]qF+F4F+F+F+%%FAILGF^rF@Q,EIS~A052505F$"#Q,&#F+FiqF+*$,&F +F+F*!#;#F+F5#F;Fiq7$-F>6,Q>A~simple~context-free~grammarF$7%FB<%/%"BG-FL6$%"CG Fbs/FB-FV6$F_sFN/Fbs-FV6%FBF_sFN%+unlabelledG77FinF+Fdp"#K"$?$"%%e$"&3I%"'s1a"( O(Gq")![;P*"+GTau7",'\#)=h<"-W\NmlC".#faA')*[$"/+c1v^&)\"0SY%4_9zr"2!G(p`0w4/"" 3?.w7j.g=:"4g8:(eK0GFA"5+KGjG`"3BG$"6g8fwjo1;y&[Far<%/F\pF+/FeoFin,&*&,&F7F+F]q "#;F+FeqF+F+*&,&F;F+F]qF;F+-Ffo6#,&F]qF+F+F+F+F+F^r,$*(%#PiG#F;F5F]q#!"$F5)FcuF ]qF+#F+FcuQ%FAILF$Q,EIS~A052704F$"$e'-F>6,QUA~simple~context-free~grammar~in~a~ labelled~universeF$7%FB<%F^sFcs/Fbs-FV6%F_sFBFNFgn77FinF+Fiq"$#>"%!o("'!3I%")gd '4$"+!)o)\s#"-?bj)R$G"/+CEOy+M"1+koJd1DY"3+GL;7**4Iq"6+/"RMC&yc5=""8+O^z#z%[[WJ <#":++s-f&epp*)GYV"<++_()\([IauT)zQ*"?++k/r*4t)R\GB,y@"A++s]>G(G&)3lYdq9S&"D++3 !*\VUerLe6,@!p$)HgDWY!f=="Far<%F^uF_u,&* &,&FiqF+F]qFerF+FeqF+F+FfuF+F^rFiuFavQ,EIS~A052734F$"$!p,$*&,&F;F+*$,&F+F+*$F*F 5!"%FfrF+F+F*F;F\v7$-F>6,Q#)Fin"0+?(QI#*4c Fin"3++Gx@-L'3&Fin"6+S=Tx=y%Q9fFin,$*&F*F;,&F+F+FhxF;F+Ffr<'F^uF_uF^p/FioF2,&*& ,(F`qF[yF\q!#7F]qF7F+FeqF+F+*&F]rF+-Ffo6#,&F]qF+F5F+F+F+F^r*(,&*(F[vF\vFdpFfrF5 Ffr#F+Fdp**-%$expG6#*(^#F+F+F[vF+F]qF+F+F[vF\vFdpFfrF5Ffr#F;FdpF+F]qF]v)Fdp,$F] qFfrF+QhnBinary~Trees~of~n~elements~(external~nodes~or~internal~nodes)F$Q,EIS~A 052510F$"#a-F>6,Q>Unlabelled~Plane~Binary~TreesF$7%FBFayFis77FinF+FinF+FinF5Fin FhpFin"#9Fin"#UFin"$K"Fin"$H%Fin"%I9Fin"%i[FinF`z<'F^uF_uF^p/FioF+,&*&F]qF+FeqF +Fdp*&,&F]qF;F^vF+F+FjzF+F+F^rF][lQepUnlabelled~Binary~Trees~of~n~elements~(ext ernal~nodes~or~internal~nodes)~with~two~branchesF$QNOdd~terms~are~the~Catalan~n umbers~EIS~A000108F$"#b,$*(F*F+,&!#>F+F*""(F+,&F;F+F*F+FboF;7#-F>6,Q4Arithmetic ~sequenceF$7%FB<#/FB-FV6$-FL6%-FV6.FNFNFNFNFNFNFNFNFNFNFNFN-%)SequenceG6#FNFd^l -FL6$-FV6)FNFNFNFNFNFNFNFd^lFis77Fin"#>"#J"#VF`]l"#n"#z"#"*"$.""$:""$F""$R""$^" "$j""$v""$(="$*>"$6#"$B#"$N#"$Z#Fa]l<&F_u,(FeqF+!"(F+F]qFhz/F\pF\_l/F_pF]_l,&Fe ]lF+F]qFjq,$F]qFjqQ*~12~n~+~7F$Q,EIS~A017605F$"$'***&,&F;F+*$F*FdpF+F+,(F;F+F[a lF+F*F+F;7$-F>6,Q:Fibonacci~like~recurrenceF$7%FB<#/FB-Fe^l6#-FL6$FN-Fe^l6#-FL6 &FNFNFNFNFis77F+F+F+F+F+F5F4FdpFhpFe]l"#5Fb\lF\_l"#E"#O"#]"#p"#&*"$J""$"="$]#Fi `l<(FdoF^u/F_pF+Fi\l/FbpF+,(FeqF+F[rF+-Ffo6#,&F]qF+FdpF+F;-%$SumG6$,$*&,*%'_alp haGFdq*$FcclF4"#[*$FcclF5"#kF_blF+F+)FcclFeuF+#F+"$$G/Fccl-F-6#,(F;F+*$F0FdpF+F 0F+*(,&F;F+*$-F-6$F^dl$"+!f>\C(!#5FdpF+F+,&FcdlF;FbdlF[yF;),&F+F+*$FcdlF4F+F]qF +FavQ,EIS~A003269F$"$x$-F>6,QSA~simple~regular~expression~in~a~labelled~univers eF$7%FBFbalFgn77F+F+F5F2"#CFhy"%g@"&g,#"'+;?"(g,a#")+!)GO"*+_$)e&"++/.,"*"-+3a- >;".+K[=%QJ"/++S=PQl"1+?F-DnV9"2++7pcI!zL"3+!o.Xb4rQ)"5+?f)*R6,Q1A~simple~grammarF$7 %FB<%/FB-Fe^l6#F_s/F_s-FL6$FbsFB/Fbs-FF6$FN1F+FSFgn77F+F+FhpF^_l"$X&"%6#*"'0`>" ($[!*\"*X".B.kG#y&)"0X\w/yQ3%"2"H^Fq.iD@"40\'3cB4*3?""5jldz Ox'3">t"7X,N7'y)4z&oy%"9JZrE"H?v.,TM$";0<"3%4???-*[`[#"=.s7fzJf^&4$)G!e>"?XtC(> +1Dy6,F`el7%FB<#/FB-F L6%FNFN-Fe^l6#-FVF[blFgn77FinFinF5Fcel"$%QF^w"'?V="(g4;&"*?2:l""+?fUXf"-+o.'\RY5"0+;s"epA]"2+K_\#=!=h#"4+#*Ht>-4EY""5+?&zR=8acx)"7+!G*ouPW'=kh&"9+S5() ymQ";+!)[ry3sGo&)z\F"=+!)3L#y'y#)*="p%)*3#"?++/Z'eUHi=&Hv(=n"Fijl<&FdflF_u F[p,&*&,&F]qF[yF[yF+F+FeqF+F+FfuF+,$*&)FdpF]qF+F_glF+F`v,$F^]mF`vFavQ,EIS~A0526 70F$"$=',$*&F*F+Ff]lFbo"#=7#-F>6,Q0Multiples~of~18F$7%FB<#/FB-FL6%-FV64FNFNFNFN FNFNFNFNFNFNFNFNFNFNFNFNFNFNFd^lFd^lFis77FinFd]mF_blF\\l"#s"#!*"$3""$E""$W""$i" "$!="$)>"$;#"$M#"$_#"$q#"$)G"$1$"$C$"$U$"$g$Fb]m<%F_u/F\pFd]m,&FeqF+F]q!#=,$F]q Fd]mFf_mFh]mQ,EIS~A008600F$"$I$,$Fc]mF57$-F>6,Q/Multiples~of~2F$7%FB<#/FB-FL6%- FV6$FNFNFd^lFd^lFis77FinF5FdpF2FiqF]blFjqFb\lFcuFd]m"#?"#AFcelF^bl"#G"#IF[t"#MF _blF`r"#SFi_m<%F_u/F\pF5,&FeqF+F]qFbo,$F]qF5F_amF]`mQ,EIS~A005843F$"$9$-F>6,F`e l7%FB<#/FB-FL6%Fd^lFd^lFc`mFgn77FinF5FiqF_blF]w"%+7"%S')"&g0("'?^k"(S=`'")+gds" *+'p"y)",+%Qg\6F[fl".+O:#*4W#"/++/J-BR"0+gTw#H&p'"2+SEbDP$47"3+!3iS`a[I#"4+ghNb "Q^AY"5++glqK!3;t*Fi_m<%F_u,&*&,(F\qF;F]qFboF;F+F+FeqF+F+*&FfuF+F]qF+F+F]am,$*& F]qF+F_glF+F5F_amFavQ,EIS~A052582F$"$E&,$*&Ff]lFbo,&F;F+F9F+F;F;7#-F>6,Q+Denume rantF$7%FB<#/FB-FL6%Fd^lFd^l-Fe^l6#-FL6%FNFNFNFis77F+F5F4FhpFe]lF:Fjq"#:Fd]mFg` mF^blFi`m"#NF[am"#X"#^"#d"#j"#q"#x"#%)Fbcm<%FdoF]am,.FeqFboFfuFboFjzFboF\qF+F]q Fe]lFjqF+,*F\q#F+F2#FiqF:F+F]q#FhpF2-F^cl6$,$Fhcl#F;F:/Fccl-F-6#,(*$F0F5F+F0F+F +F+F+,$F\qF_emQYnumber~of~ways~to~make~n~cents~with~coins~of~1~1~3~centsF$Q,EIS ~A001840F$"$2#*&,(-F_jl6#,$*$Ff]lF;F;F;*&FafmF+F*F+F+F*F;F+Ff]lF;7#-F>6,Fav7%FB <%/FB-FV6$F_sFbs/F_s-FIFf^l/Fbs-FF6$F_sFjhlFgn77FinF5F4FiqFi`mFe^m"$S)"%gd"&g`% "'+KS"(!o"*R")+caV"*+%=*=&"++C-1n",+?J0M*".+#fE&[R""/+gDk/BA"0+S)z@5mP"1+S#Q8h! en"3++c9TZZ!G""4+?Z&e3raaDF_fm<&F_u/F_pF4F]am,&*&,(F\qF;F]qF[yF^vF+F+FfuF+F+*&F \[lF+FjzF+F+F^rF+FavQ,EIS~A001048F$"$,",$*$FafmF5Ffr7#-F>6,Q+Cycles~SetF$7%FB<# /FB-FF6$F_gmFRFgn77FinFinF+F4"#6F`bl"$u#"%k<"&oI""'%e4""(wl-"")S'G1""*SQa?""+!) GW'["",S!fF!)>"-gtklMG".+;+j"RV"/+OR#GM2("1+#z0f0MA""2+;_e!))pPA"3+%Qw"o9l:VF[i m<&F_uF[pFgbl,(*&,(F\qF+F]qF5F+F+F+FfuF+F+*&,&F]qFboF^vF+F+FjzF+F+F[rF+F^r*&-F_ jlFfqF+F]qF;Q^oPermutation~with~exactly~two~cycles,~Stirling~numbers~of~first~k indF$Q,EIS~A000254F$F]_l*&,(F+F+F*FboFjxF+F+,(F+F+F*F^vFjxF+F;7#-F>6,Fav7%FB<&/ F_sFd^l/Fbs-Fe^lFihl/FB-Fe^l6#%"EG/F_\n-FLF]gmFis77F+F+F4Fiq"#@F`]lFe^mF]el"$() *"%%e#"%ln"&6x""&oj%"'$R@""'6yJ"'S?$)"(4$y@"(()Gq&")_.$\"")p")3R"*bTL-"Fa[n<&Fd oF^u,(FeqF+FfuF^vFjzF+Fdhm-F^cl6$,$*&,&FcclF4FboF+F+FhclF+#F;Fhp/Fccl-F-6#,(Fje mF+F0F^vF+F+*(,(FboF+*$FhpFfrF+*$,&#F4F5F+F_^nF\vF5F+F+,(#F:F5F+F_^nF]vF`^nFboF ;)Fa^nFfjlF+FavQ,EIS~A001906F$"$Z",$*(F*F+,&FgdlF+F*F+F+Ff]lFboF;7#-F>6,Fj]l7%F B<#/FB-FV6$-FL6%-FV6+FNFNFNFNFNFNFNFNFNFd^lFd^l-FL6$-FVFf^lFd^lFis77FinF]blF\_l Fh`m"#P"#YF`]lFgcl"#t"##)Fa_l"$+""$4""$="Fd_l"$O""$X""$a"Fg_l"$s"FdblFh^n<&,(Fe qF+F;F+F]qFdqF_u/F_pF\_l/F\pF]bl,&F+F+F]qF:,$F]qF:Q)~9~n~+~1F$Q,EIS~A017173F$"$ j*,$*(,(FboF+F*F+FjxF5F+,&F;F+FjxF5F;Ff]lF;F;7$-F>6,Q6,F`el7%FBFhanFgn77F5F+F2F2"$?"Fcen"%!['"%S]"'Sao"'!)GO"*+/v>"")+o"*R" ,+S5N6$"++3-Fi"/+[c**fC6".+!oVn28"1+g@,q:x`"0+g4Guob$"4+SYQ5r6,F`el7%FB<#/FB-FV6$Fd^lF`dmFgn77F+F+F5FjqFcelFcen"$?(F een"&?.%Fgen"(+)GOFien"*+;+z%F[fn",+7Hyr)F]fn"/+!)))*yA4#F_fn"1+!GdqtBS'Fafn"4+ +kw"3?!HV#F]gn<(FdoF^uFdflF]gl,&*&FeuF+FeqF+F+FfuF+/FioFjqF_glF+FavQ,EIS~A05256 5F$"$2&,$*(F*F+,&F[yF+F*F+F+Ff]lFboF[y7#-F>6,Fj]l7%FB<#/FB-FV6$F`^l-FL6$Ff[mFd^ lFis77FinFcuFh`mF[am"#_Fgcl"#w"#))F_`n"$7""$C"Fb`n"$[""$g"Fe`n"$%="$'>"$3#"$?#" $K#"$W#Ffhn<&F_u,(FeqF+F[yF+F]qFhz/F\pFcu/F_pFh`m,&FdpF+F]qFjqFe`lQ*~12~n~+~4F$ Q,EIS~A017569F$"$$**,$*&Ff]lF+,,F+F+F*F;F[alFbo*$F*FhpF5F9F;F;F;7#-F>6,Ffan7%FB <#/FB-Fe^l6#-FL6&-FV6%Fd^lFNFNFNFNFNFis77F+FinFinF+F4F+F5Fe]lFjqF]blF\_l"#T"#hF ein"$N"Fhy"$c$"$@&"$z)"%X9"%)>#Fijn<(FdoF[pF^pFi\l/FbpF4,,FeqF5FfuFboFjzF;FjblF ;-Ffo6#,&F]qF+FhpF+F+-F^cl6$,$*&,,!$=%F+Fccl!%4ZFfcl"$*fFdcl"%[5*$FcclFdp"$U&F+ FhclF+#F;"&8*>/Fccl-F-6#,,F+F+F0F;F_dlFbo*$F0FhpF5*$F0F4F;,$*(,&F;F+-F-6$Fh]o$" +?_"HJ'FgdlF+F+,**$F^^oFhpFgdlF^^oF+*$F^^oF4F4*$F^^oFdpFiqF;),*F+F+*$F^^oF5F+Fd ^oF5Fe^oFboF]qF+F;FavQ,EIS~A052914F$"$'*),$*&Ff]lFbo,&F;F+*$F*FiqF+F;F;7#-F>6,F hcm7%FB<#/FB-FL6%Fd^lFd^l-Fe^l6#-FL6*FNFNFNFNFNFNFNFNFis77F+F5F4FdpFhpF2Fe]lFiq F]blFjqFb\lFcuFd]mFf`mFg`mFcel"#FFi`mFbbnF_bl"#RF[_o<*Fdo/FbpFhpFdhmF]am/FioFdp ,8FeqFboFfuFboFjzFboF[rFboFjblFboFe\oFbo-Ffo6#,&F]qF+F2F+Fbo-Ffo6#,&F]qF+Fe]lF+ FboF\qF+F]qFabnFa^mF+/-Ffo6#F2Fe]l/FfpF2,*F\qF`v#F]`oF[tF+F]q#FhpFiq-F^cl6$,$*& ,.FfclFhpFdclFiqFa]oF:*$FcclFhpFiq*$FcclF2FhpFa`lF+F+FhclF+#F+"$G"/Fccl-F-6#,2* $F0Fe]lF+*$F0F2F+Fi]oF+F_dlF+Fj]oF+FjemF+F0F+F+F+F+,$F\qF`vQYnumber~of~ways~to~ make~n~cents~with~coins~of~1~1~8~centsF$Q,EIS~A008726F$"$">*&Ff]lF;,(F;F+F*F+F9 F+F;7#-F>6,Ffan7%FB<#/FB-FL6$Fd^l-Fe^l6#-FV6$FNF`dmFis77F+F5F4FhpFiqFjqFd]mF\`o F[am"#f"#()Fgao"$)="$w#"$0%"$%fFhjl"%x7"%s="%WF"%ASFbbo<&FdoFdhmF]am,*FeqF;F;F+ FjzF;F[rF+,&F;F+-F^cl6$,$*&,("#BF+FfclF\_lFccl"#8F+FhclF+#F+F]_l/Fccl-F-6#,(F;F +F0F+Fj]oF+F+*(,&F;F+-F-6$Fido$"+Q!yK#oFgdlF+F;,&F\eoF;*$F\eoF4F^vF;),&F+F+*$F\ eoF5F+F]qF+FavQ,EIS~A020745F$"$x)*(F*F+,(F*FboFjxFdpF;F+F+,&F;F+F*F5F;7#-F>6,F` el7%FB<#/FB-FL6$FN-FV6%FNFN-Fe^l6#Fc`mFgn77FinF+FiqFcelF]w"%?>"&SI#"'gDKFj[m")! G(*G*"++c%z&=",+K![(3%"-+oF&*4)*"/+o>xe]D"0+/^hX;9("2+?JXo$\U@"3+S)*\!zzf&o"5+g X*p(GJ.JB"6+gT!=dj#>6,Q,HierarchiesF$7%FB<*/&%"AGF]p-FF6$&F]joF`pFjh l/F`jo-FF6$&F]joFjoFjhl/FB-FF6$F\joFjhl/Fdjo-FF6$&F]joFcpFjhl/F[[p-FF6$&F]joFgp Fjhl/F_[p-FF6$&F]joFj`oFjhl/Fc[p-FF6$&F]jo6#Fe]lFjhl/Fg[pFNFgn77FinF+Fe]lFidm"$ 5*"&KX""'yZF"(#*e*f"*g[:[""+A'>c3%"-]!HYIC"".!\(HnQ8%"0gGg?T6\""1Y\mfMV'z&"3\LG $*QN#[T#"5fy^`H&*H1t5"6A^dmV6ZL8";_`wB1_Fs$QOBR ("=)Ru%yPyg)=3i3(*H%FchoF^rF^rF^rQ8Hierarchies~of~height~7F$Q,EIS~A001669F$"$'H ,$*8,6FgdlF+F*"$&\F9"']q7Fjx!&g0"*$F*Fe]l")+&G_#F^_o!)_r]DF[al!'Qm%*F\[o"(v-^%* $F*F2!)?xm8*$F*F:F]jmF+F[[mF;Ff]lF;FieoF;,&F;F+F*F4F;,&F;F+F*FiqF;,&F;F+F*FhpF; ,&F;F+F*F2F;,&F;F+F*Fe]lF;,&F;F+F*F:F;,&F;F+F*F]blF;F;7#-F>6,Q&ListsF$7%FB<#/FB -FV6,Fd^lFdfo-Fe^l6#-FVFadmFd[m-Fe^l6#-FV6'FNFNFNFNFN-Fe^l6#-FV6(FNFNFNFNFNFN-F e^l6#Fi^l-Fe^l6#-FVFj_o-Fe^l6#Fe_n-Fe^l6#-FV6,FNFNFNFNFNFNFNFNFNFNFis77F]blF`]l "$&Q"%DI"&L`#"'D3A"(0%y>")D/3="*L8tn""+&)\Iu:",D>M9\""-D'>VOU"".Lh`GuO""/X6wgG? 8"0XM&H!y.G""1D'zj&[KY7"2L(4?812<7"30tPN*=$z">""4l\[l6*G%)p6"5DCD,-^%*p]6"6L@bK )e=8%R8"F_]p<-/-Ffo6#FiqFj`p/-Ffo6#F:F[ap/-FfoFh[pFi`p/Fi`oFh`p,8Feq!(+)GOFfu"( S)**pFjz!(OPv&F[r"(kdl#Fjbl!'m6wFe\o"'*)39Fb`o!&%)o"Fe`o"%m7-Ffo6#,&F]qF+FiqF+F aq-Ffo6#,&F]qF+F:F+F+FgenF+/FbpFf`p/FfpFg`p/FioFe`p/F\pF`]l/F_pFd`p/FeoF]bl,6F^ ]mF+F+F+F`hoF+)F4F]qF+)FiqF]qF+)FhpF]qF+)F2F]qF+)Fe]lF]qF+)F:F]qF+)F]blF]qF+F]d pQ2Sum~of~nth~powersF$Q,EIS~A001557F$"$q$-F^cl6$*(-&%*numtheoryG6#%$phiG6#&%"jG F]pF+FjdpF;-F_jl6#,$*$,&F;F+)F*FjdpFhpF;F;F+/Fjdp;F+%)infinityG7#-F>6,Q7Necklac es~of~5~coloursF$7%FB<(/FB-FI6#-FV6'%#c1G%#c2G%#c3G%#c4G%#c5G/Fdfp%%AtomG/F`fpF ffp/FafpFffp/FbfpFffp/FcfpFffpFis77FinFhpFcdmFedm"$l""$H'"%NE"&l6""&:*["'Xq@"'( )o(*"(D*QW")&[Y.#")X-!R*"*&*4(fV"+hc]M?"+lwwO&*",l8zy[%"-NhtF>@".&[,x'Q+"".d22s $oZFadpF^rF^rF^rFhepQ,EIS~A001869F$Fhp*0Ff]lFbo,&F;F+FjxF+F;,&F;F+Fj]pF+F;,&F;F +*$F*FjqF+F;,&F;F+*$F*FcelF+F;,&F;F+*$F*FeclF+F;,&F;F+*$F*"#gF+F;7#-F>6,Fhcm7%F B<#/FB-FL6*Fd^lFd^l-Fe^l6#-FLFd`m-Fe^l6#-FLFg_p-Fe^l6#-FLFc^l-Fe^l6#-FL6:FNFNFN FNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFN-Fe^l6#-FL6RFNFNFNFNFNFNFNFNFNFNFNFNF NFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFN-Fe^l6#- FL6hnFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFN FNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFis77F+F5FdpF2F:FjqFabnFg`m"#HF_b lFedmF\\lF__l"#!)"#(*"$9"F\\o"$c""$$="$5#"$V#F^hp<_tFdoFdz/F_pFdpF]am/-Ffo6#"#u "&Y*>/-Ffo6#"#v"&[5#/-Ffo6#Fein"&[A#/-Ffo6#Fjdm"&[M#/-Ffo6#"#y"&oZ#/-Ffo6#F`_l" &)3E/-Ffo6#Fijp"&Gv#/-Ffo6#"#")"&o*G/-Ffo6#F^`n"&G0$/-Ffo6#"#$)"&)3K/-Ffo6#F[em "&)zL/-Ffo6#"#&)"&3b$/-Ffo6#"#')"&ot$/-Ffo6#Fbco"&G#R/-Ffo6#Ffin"&Q7%/-Ffo6#"#* )"&[K%/-Ffo6#Fb^m"&Qa%/-Ffo6#Fa_l"&Gw%/-Ffo6#"##*"&)**\/-Ffo6#"#$*"&oB&/-Ffo6#" #%*"&=\&/-Ffo6#Fbbl"&ou&/-Ffo6#"#'*"&R-'/-Ffo6#Fjjp"&5I'/-Ffo6#"#)*"&-g'/-Ffo6# "#**"&%**o/-Ffo6#F_`n"&2A(/-Ffo6#Fjhm"&?a(/-Ffo6#"$-""&&*)y/-Ffo6#Fb_l"&qB)/-Ff o6#"$/""&2h)/-Ffo6#"$0""&W)*)/-Ffo6#"$1""&VQ*/-Ffo6#"$2""&Uy*/-Ffo6#Fc^m"'d@5/- Ffo6#F``n"'sk5/-Ffo6#"$5""'.66/-Ffo6#"$6""'Md6/-Ffo6#Fgin"'"o?"/-Ffo6#"$8""'Gc7 /-Ffo6#F[[q"'X48/-Ffo6#Fc_l"'ii8/-Ffo6#"$;""'\>9/-Ffo6#"$<""'Ow9/-Ffo6#Fa`n"'$p `"/-Ffo6#"$>""'](f"/-Ffo6#Fcen"'[i;/-Ffo6#"$@""'YF/-Ffo6#"$D""'%Q,#/-Ffo6#Fd^m"'w#4#/-Ffo6#Fd_l"'or@/-Ffo6#F gao"'sbA/-Ffo6#Fdbn"'wRB/-Ffo6#"$I""'#*GC/-Ffo6#Fcbl"'3=D/-Ffo6#Fd\l"'E8E/-Ffo6 #"$L""'W3F/-Ffo6#"$M""'k4G/-Ffo6#Fb`n"'1=I/-Ffo6#F\\o"'%3"H/-Ffo6#"$P""'GDJ/-Ff o6#"$Q""'URK/-Ffo6#Fe_l"'c`L/-Ffo6#"$S""'iuM/-Ffo6#"$T""'o&f$/-Ffo6#"$U""'mBP/- Ffo6#"$V""'k^Q/-Ffo6#Fe^m"'o()R/-Ffo6#Fc`n"'sBT/FfpFjq/Fi`oFabn/F_bpFg`m/FiapFh jp/F\bpF_bl/-Ffo6#F]blFedm/-Ffo6#FgimF\\l/-Ffo6#FjqF__l/-Ffo6#FddoFijp/-Ffo6#Fb \lFjjp/-Ffo6#FcdmF[[q/-Ffo6#FcuF\\o/-Ffo6#FabnF\[q/-Ffo6#Fd]mF][q/-Ffo6#F\_lF^[ q/-Ffo6#Ff`mF_[q/-Ffo6#Fc\nFdco/-Ffo6#Fg`m"$:$/-Ffo6#Fcdo"$a$/-Ffo6#Fcel"$.%/-F fo6#"#D"$_%/-Ffo6#F^bl"$6&/-Ffo6#F\`o"$q&/-Ffo6#Fh`m"$R'/-Ffo6#Fhjp"$3(/-Ffo6#F i`m"$"z/-Ffo6#F]_l"$u)/-Ffo6#F[t"$r*/-Ffo6#Fbbn"%o5/-Ffo6#Fj`m"%z6/-Ffo6#Fddm"% !H"/-Ffo6#F_bl"%@9/-Ffo6#F[`n"%_:/-Ffo6#F`r"%.#/-Ffo6#Fc\l"%$R#/-Ffo6#F^_l"%!f#/-Ffo6#"#W"%8G/-Ffo6#Fedm"%OI/-Ff o6#F\`n"%&G$/-Ffo6#"#Z"%MN/-Ffo6#Fecl"%>Q/-Ffo6#"#\"%/T/-Ffo6#F`bl"%DW/FbpF:/-F fo6#Ffdm"%YZ/-Ffo6#Fdin"%.^/-Ffo6#"#`"%ga/-Ffo6#F\\l"%je/-Ffo6#F`]l"%mi/-Ffo6#" #c"%:n/-Ffo6#Fgdm"%kr/-Ffo6#"#e"%fw/-Ffo6#Faco"%a")/-Ffo6#Fihp"%5()/-Ffo6#F[\o" %m#*/-Ffo6#"#i"%$))*/-Ffo6#Fhdm"&+0"/-Ffo6#Fgcl"&y6"/-Ffo6#Fcbn"&c="/-Ffo6#"#m" &5E"/-Ffo6#F__l"&kL"/-Ffo6#"#o"&%>9/-Ffo6#Fabl"&C]"/-Ffo6#Fidm"&If"/-Ffo6#"#r"& Oo"/-Ffo6#Fa^m"&Sy"/-Ffo6#F]`n"&W)=,b^l!1+'p.zIvq"F+F]q!/ku!3%*)pz*$F]qF2!%]5Fe qFeenFfu"&S-$Fjz"'+35Fcq!*+X6="Fe\o"'Sy)*F[r"'+?DFjbl"'+#H&Fb`o"(+%)o"F\q!.]?-: Uf"F`q!,\+g:x"*$F]qFhp!''[s%-Ffo6#,&F]qF+F]hrF+"*+gXN%-Ffo6#,&F]qF+FedmF+"*SU(G X-Ffo6#,&F]qF+F^_lF+"*gx.=%-Ffo6#,&F]qF+Fj[oF+"*!G,KQ-Ffo6#,&F]qF+Fc\lF+"*?&>1S -Ffo6#,&F]qF+F`rF+"*+;&4L-Ffo6#,&F]qF+F]`oF+"*+[O[$-Ffo6#,&F]qF+F[amF+"*SIyl$-F fo6#,&F]qF+F]^sF+"*!oxww-Ffo6#,&F]qF+FcbnF+"*![!ee(-Ffo6#,&F]qF+FhdmF+"*?NaP(-F fo6#,&F]qF+FgclF+"*+c^[(-Ffo6#,&F]qF+F\]sF+"*?ZsD(-Ffo6#,&F]qF+FihpF+"*!)))y*p- Ffo6#,&F]qF+F[\oF+"*!o>Jr-Ffo6#,&F]qF+F[\sF+"*?>>r'-Ffo6#,&F]qF+FacoF+"*+Gz&o-F fo6#,&F]qF+Fb[sF+"*!o-Ffo6#,&F]qF+ F^blF+"*+;GO"-Ffo6#,&F]qF+F\`oF+"*?FG]"-Ffo6#,&F]qF+F`crF+"*+c#H7-Ffo6#,&F]qF+F cdoF+")++G)*-Ffo6#,&F]qF+FcelF+"*++D5"-Ffo6#,&F]qF+Fg`mF+")?2.()-Ffo6#,&F]qF+Fc \nF+")+s]w-Ffo6#,&F]qF+F_blF+"*!)3@'H-Ffo6#,&F]qF+FddmF+"*SQ%*y#-Ffo6#,&F]qF+Fj `mF+"*+G!=E-Ffo6#,&F]qF+F[tF+"*?b7G#-Ffo6#,&F]qF+FbbnF+"*+K%[C-Ffo6#,&F]qF+Fh`m F+"*+O)[;-Ffo6#,&F]qF+F[`nF+"*g&eNJ*$F]qFe]lF;-Ffo6#,&F]qF+FjqF+")?RN9-Ffo6#,&F ]qF+FddoF+")ghT=-Ffo6#,&F]qF+FgimF+")+o$4"-Ffo6#,&F]qF+F]blF+"(S%>")F]cp"(+k%eF jbp"(+s0%Fe`o"(g8p#-Ffo6#,&F]qF+Fb\lF+")+)eJ#-Ffo6#,&F]qF+Fc_sF+"*!oJlz-Ffo6#,& F]qF+Fa^mF+"*+Gc)z-Ffo6#,&F]qF+FidmF+"*!3!=$z-Ffo6#,&F]qF+FablF+"*+%e&)y-Ffo6#, &F]qF+Ff^sF+"*?rs#y-Ffo6#,&F]qF+F__lF+"*?nuv(-Ffo6#,&F]qF+FijpF+F`cs-Ffo6#,&F]q F+Fg\qF+Fh`t-Ffo6#,&F]qF+F`_lF+F\at-Ffo6#,&F]qF+FeinF+F``t-Ffo6#,&F]qF+FjdmF+Fd `t-Ffo6#,&F]qF+Fd]qF+Fdcs-Ffo6#,&F]qF+F^`nF+F\ds-Ffo6#,&F]qF+F]^qF+Fhcs-Ffo6#,& F]qF+FbcoF+F`es-Ffo6#,&F]qF+FjhqF+F`[t-Ffo6#,&F]qF+Ff^qF+Fhds-Ffo6#,&F]qF+F[_qF +Fdds-Ffo6#,&F]qF+F[emF+F`ds-Ffo6#,&F]qF+F_iqF+Fh[t-Ffo6#,&F]qF+Fd^mF+Fhhs-Ffo6 #,&F]qF+F]jqF+F`\t-Ffo6#,&F]qF+FhinF+F\\t-Ffo6#,&F]qF+FdiqF+Fd[t-Ffo6#,&F]qF+Fc gqF+Fdjs-Ffo6#,&F]qF+Fa`nF+Fh]t-Ffo6#,&F]qF+FhgqF+F`js-Ffo6#,&F]qF+Fc_lF+F\js-F fo6#,&F]qF+F[[qF+F`]t-Ffo6#,&F]qF+F\dqF+F`bs-Ffo6#,&F]qF+Fb_lF+Fhas-Ffo6#,&F]qF +FcenF+Fhjs-Ffo6#,&F]qF+FahqF+F\[t-Ffo6#,&F]qF+FadqF+F\bs-Ffo6#,&F]qF+F``nF+F\^ t-Ffo6#,&F]qF+Fc^mF+Fdbs-Ffo6#,&F]qF+F[eqF+Fhbs-Ffo6#,&F]qF+FfdqF+F\cs-Ffo6#,&F ]qF+FginF+F\]t-Ffo6#,&F]qF+FabqF+Fhgs-Ffo6#,&F]qF+FffqF+Fd]t-Ffo6#,&F]qF+F]fqF+ Fh\t-Ffo6#,&F]qF+FheqF+Fd\t-Ffo6#,&F]qF+FccqF+F`as-Ffo6#,&F]qF+FjhmF+Fdas-Ffo6# ,&F]qF+F_`nF+F\hs-Ffo6#,&F]qF+FfbqF+F`hs-Ffo6#,&F]qF+Fh_qF+Fhes-Ffo6#,&F]qF+Fj` qF+F\gs-Ffo6#,&F]qF+Fe`qF+F\fs-Ffo6#,&F]qF+Fa_lF+F`fs-Ffo6#,&F]qF+Fb^mF+Fdes-Ff o6#,&F]qF+FhaqF+F`gs-Ffo6#,&F]qF+FfinF+F\es-Ffo6#,&F]qF+FjjpF+Fdgs-Ffo6#,&F]qF+ FbblF+Fdfs-Ffo6#,&F]qF+F_aqF+Fhfs-Ffo6#,&F]qF+Fe[qF+F\`t-Ffo6#,&F]qF+Fj[qF+Fh_t -Ffo6#,&F]qF+F]`nF+"*+KC*z-Ffo6#,&F]qF+FcblF+F`is-Ffo6#,&F]qF+Fb[rF+Fhis-Ffo6#, &F]qF+FdbnF+Fdis-Ffo6#,&F]qF+FgaoF+Fdhs-Ffo6#,&F]qF+Fd_lF+F\is-Ffo6#,&F]qF+Fe_l F+F`_t-Ffo6#,&F]qF+Ff]rF+F__t-Ffo6#,&F]qF+Fa]rF+F^_t-Ffo6#,&F]qF+Fb`nF+F]_t-Ffo 6#,&F]qF+F\\oF+Fi^t-Ffo6#,&F]qF+Fd\rF+Fa^t-Ffo6#,&F]qF+F_\rF+Fe^t-Ffo6#,&F]qF+F d\lF+Fd_t-Ffo6#,&F]qF+"$Y"F+Feen-Ffo6#,&F]qF+Fc`nF+Fb`s-Ffo6#,&F]qF+Fe^mF+Fc`s- Ffo6#,&F]qF+F^_rF+Ff`s-Ffo6#,&F]qF+Fi^rF+Fg`s-Ffo6#,&F]qF+Fd^rF+Fe`s-Ffo6#,&F]q F+F_^rF+Fh`s,XF]q#",8:LsE"",)[([R?"F``s#Fjdm"++;OmrFcq#"'&GJ""*G*3LdF\q#"+"y*zt 6"++WdxZF`q#"*,hsk"",+KsKV"F[as#"&Z2"Fh_uF]^t#F+",+7`k,&)F;Ffjl#")@lp@"*+s%p<*& F``uF+F]qF5#"&>H("*?$oh5*&F``uF+F]qFdp#Fjdm"*!)[^b**&F``uF+F]qFhp#F+Fh_u*&F``uF +F]qF4#"&RA$"+SYamG*&F``uF+F]qF+#")"f&Q\F`au-F^cl6$,$*(,*"((*yd"F+Fccl"(;&\[Fdc l"(=Z0#Ffcl"()p:CF+)Fccl,&FboF+F]qF;F+FhuF+#F;"+SU&\2"/Fccl-F-6#,(F_dlF+FjemF+F +F+F+-F^cl6$,$*.,*FcclF5FdclF+FfclFdpF5F+F+)Fccl,&!"&F+F]qF;F+FhuF+F\[lF+F]rF+F \clF+#F+F`buFabuF+-F^cl6$,$*,,,Fa]oFdpFdaoFhpFdclF4FcclF^vF[yF+F+)FcclF\glF+Fhu F+F\[lF+F]rF+#F;")?J")R/Fccl-F-6#,&F]boF+F+F+F+-F^cl6$,$*&,:Fa]o"$C'Fccl!#XFeao "%35*$FcclF:"$R**$FcclFe]l"%P5Fdcl"$6%Fdao"$>)*$FcclFgim"$\&FfclF]w*$FcclFiq"%k 5*$FcclF]bl"$;)!$)GF+F+FhclF+#F;"&K%=/Fccl-F-6#,&*$F0FjqF+F+F+F+-F^cl6$,$*&,PFc cl!#HFfcl!#MFdcl!#QFa]o!#UFdaoF`duFeao!#[Fddu!#\FjduFiflFbduF`fuF\euF_fuFhduF`d u*$FcclFjqF^fu*$FcclFddoF]fu*$FcclFb\lF\fu*$FcclFcdmF[fu*$FcclFcuFjfl*$FcclFabn Fe_m*$FcclFd]mFhz*$FcclF\_lF1*$FcclFc\nF2*$FcclFg`mFjq*$FcclFcdoFd]mFjflF+F+Fhc lF+#F;Fh[m/Fccl-F-6#,&*$F0FcelF+F+F+F+-F^cl6$,$**,*Fccl"&\d$Fdcl"&)pgFfcl"&eh'! &e`&F+F+)FcclF]]lF+FhuF+F\[lF+F_buFabuF+-F^cl6$,$*,,*Fccl!$'GFdcl"$5$Ffcl!$P"!$ )HF+F+FccuF+FhuF+F\[lF+F]rF+#F;"*?rZP&FabuF+-F^cl6$,$*&,.Ffcl!(W"3:Fa]o!')[N&Fd ao"'r/UFccl!(hlS"Fdcl!(F<+"!(SIG"F+F+FhclF+#F+FecuFfcuF+-F^cl6$,$*(,8FdaoF2Fa]o F4FbduFb\lF\euFddoFeaoF:FdduFjqFjduFddoFfclF^vFhduFjqFcclF1FdqF+F+F]buF+FhuF+F_ euFaeuF+#!,RF`Tf""+%yE*>6,F^hl7%FB<$ F^gm/FB-FL6(FNFNF_sF_sF_sF_sFgn77FinFinFinFinFinFinFhgn"&!35"'SU6"(!3q7")S5i9"* g(Hw<"+S#=EH#",+!o^[J"-ofFn(f%".?b.Nm7("0gH$y@2q6"1+7*=*y8I?"2cM))[kdQr$"3K+UtF ;MZr"5g0pM'eM#)RW"F]`v<*F_uF[pF^pFap/FioFinFep/Fi`oFhgn,,*&,4Fh[mF+F``s!#@F]qF[ tF`qFeclFcqFjinF[as!#AF\q!$k%F]^tF5*$F]qFiqF+F+FeqF+F+*&,2!$g$F+F]q"$U'F\q!$@"F cqFadqF`q!$'HF[asFeclF]^tF[yF``s!#9F+FfuF+F+*&,.F`q!#!*F]q!#%)FcqFhflF``sF2F[as FcelF\q"$z"F+FjzF+F+*&,,F]qF7F`qFjqFcqFebvF\qFb\lF[asF[yF+F[rF+F+*&,*FcqF+F\qF; F`qF5F]qFboF+FjblF+F+F^r,$*&F^[nF4F]qF;FdpFavQ,EIS~A052790F$"$Z(,$*(F*F+,&F^vF+ F*F+F+Ff]lFboF[y7#-F>6,Fj]l7%FB<#/FB-FV6$-FL6%F\`pFd^lFd^lFainFis77FinFjqFf`mFh `mF_blF]hrFdinFihpFf^sFeinF[emFe`qF_`nFc^mFcgqFhinFd\lF_^rFiinF\[q"$k"Fccv<&F_u /F_pFf`m/F\pFjq,(FeqF+F[yF+F]qF7,&FdpF+F]qFiq,$F]qFiqQ)~8~n~+~4F$Q,EIS~A017113F $"$f**(F*F+Ff]lF+F]^pF;7#-F>6,F`el7%FB<#/FB-FL6$FN-Fe^l6#-FL6$Fd^lFc`mFgn77FinF +FdpF_blFh]vFden"'Sm6"(S%\C")glye"+?rB(e"",+O61A" 2+%Qs]O-m#*"4+!Gd#Gk5(pT"6+S%\jb36Y,?"8+S9#QPOl^u?5"9+gxN'=kH*Q-7b";+?BRA'e*)*= O&=9$"=++#RNM6,Q0k -gonal~numbersF$7%FB<&/F_s-FL6%FbsFbsFbs/F_\n-FL6$FNF_s/FB-FV6&F_sF_\nF_\nF_\n/ FbsFd^lFis77F+F2FcdmFh`mFedmF]^sFa_lFcenFf\v"$!>"$J#Fdco"$D$"$y$"$N%"$'\"$h&"$I '"$.("$!y"$h)Fbgv<%Fdo/F\pF2,*FeqF;F\qF5F+F+F]qF4,(F\qF5F+F+F]qF4,$F\qF5Q1~3-go nal~numbersF$Q,EIS~A000384F$"$S$,$*$FcboF;F;7#-F>6,F`el7%FB<#/FBF\coFgn77F+F+F5 FjqFa^m"$!["%?VFfgm"'gT_"(?Z*o"*+kg,""++))eO;",+g4S(G"-+/$y(zaF\fn"0+?bb/:Z#"1+ g(*)z7cz&"3+g(p!e44W9"4++;3\N7%4Q"6+S],lbFX21""7++#fu\kw[#4JFiiv<&FdoF^uFdfl,(* &,*F]q!#6F1F+F`qF;F\qF1F+FeqF+F+F\]lF+F[rF+*&-F^cl6$,$*&,(FdpF+FfclF2FcclF:F+Fh clF+FedoFfdoF+F_glF+,$*&F`eoF;FbeoF+F;FavQ,EIS~A052556F$"$(\*&,(F*F;-Fc[l6#-F^c l6$**)F;FjdpF+FaepF+,&FaepF+F;F+F;FjdpF;FbepF;*&Fa\wF+F*F+F+F+Ff]lF;7#-F>6,F^hl 7%FB<%F[gmFj[n/F_s-%)PowerSetG6#FbsFis77F+F5F5F4F4FdpFhpF2Fe]lF:FgimFddoFcuF\_l FcdoFh`mFbbnF]`oFjhrF`]lFcbnF_\wF^rF^rF^rFavQ,EIS~A052839F$"$1)*&Ff]lF+,(F;F+F* F+FjxF5F;7$-F>6,Ffan7%FB<#/FB-Fe^l6#-FL6%FNFd^lFc`mFis77F+FinF5F5F2F]blFg`mFc\l F[_q"$q"F`_m"$#o"%m8"%IF"%ia"&A4""&Y=#"&!pV"&#Q()"'iZ<"'E&\$Fe]w<%FdoF[p,(FeqF5 FfuF+FjzF;-F^cl6$,$*&,&F;F+FcclFhpF+FhclF+#F+F:/Fccl-F-6#,(F;F+F0F+FjemF5,$F`ho #F+F4FavQ,EIS~A014113F$"$d%-F>6,F`el7%FBF[^wFgn77F+FinFdpFjqFe^mFjam"&Se""'!o6# "(?vY$")+'*oh"++'\5C"",+wDBs#"-+c=;Vl"/+Sym(**p""0+W`Ey;w%"2+gHZ%>CG9"3+![K*yEz qX"5++C9Nt$)*Rb""6+g4CR:>AXf&"8+S)zH[w.T*e7#"9++ks#[*4t]i.&)Fe]w<%,(*&,(F\qFboF [yF+F]qF1F+FeqF+F+*&F^buF+FfuF+F+FjzF+FdoF[p*&F__wF+F_glF+Fi_wFavQ,EIS~A052598F $"$V&,$*(,&F*F+F5F+F+F*F+F_hpF;Fbo7#-F>6,Ffan7%FB<#/FB-FL6$-FV6$FbipFd^lFc`mFis 77FinFdpF5FdpF5FdpF5FdpF5FdpF5FdpF5FdpF5FdpF5FdpF5FdpF5Fhaw<&FdflF_u,(FeqF;FfuF ;F2F+/F\pFdp-F^cl6$*&,&F5F+FcclF+F+FhclF+/Fccl-F-6#,&F;F+FjemF+,&F4F+Fb[lF+FavQ ,EIS~A010694F$"%t5-F^cl6$*(FddpF+FjdpF;-F_jl6#,$*&Fg\wF5,&F;F+FaepF5F;F;F+Fbep7 #-F>6,F^hl7%FB<%/FB-FIFa]w/F_sF[\n/Fbs-FL6$F_sF_sFis77FinFinF+F5FdpF2FjqFd]mFj` mF[\sFfdq"$'="$]$F]iv"%!=""%!>#"%9T"%5x"&+Y""&%fF"&'[_FdcwF^rF^rF^rQ=Cycles~of~ pairs~of~SequencesF$Q,EIS~A052823F$"$)y,$*&Ff]lF5,*F;F+F*FdpFjxF^vF9F+F;F;7$-F> 6,Ffan7%FB<#/FB-Fe^l6#-FV6$FN-FL6%FNFd^lFd^lFis77F+F5F2F\_lFihp"$*="$&f"%t="%'* e"&g&="&D%e"';R="'\*y&"(tC#="(hpt&")u$f!=")'3\o&"*$=b*y""*[GLj&"+H\Jt<"+bj@#e&F dew<&FdoFdfnF]am,*FeqF+FfuF^vFjzFdpF[rF;-F^cl6$,$*&,(F[yF+FcclFa`lFfclF5F+FhclF +#F;F]_l/Fccl-F-6#,*F;F+F0FdpFjemF^vFj]oF+,$*(,&F;F+-F-6$F_hw$"+i>swJFgdlF+F5,( *$FchwF5F2FchwF[y*$FchwF4F^vF;),(FdpF+FchwF^vFhhwF+F]qF+F;FavQ,EIS~A052544F$Fbj v-F>6,F`el7%FBF[fwFgn77F+F5FjqF[[q"%S9"&!oA"'+%G%"(?*R%*"*?nsP#"++G0Nn"-++k7?@" .+))=Q8M("0+%=t\[!y:O#"7+!oB%Gq"[T%*=""8+!o:-> dy3@lj":+SM`"o!3OTnmg$"<+!GH0*y&o%fs]r:#">++?Zp`$)f:!Q&)4e8Fdew<&FdoF]am/F_pFjq ,*Fb[wF+*&,(F\qF4F]qFcdmFd]mF+F+FfuF+F+*&,&FhzF+F]qF[yF+FjzF+F+F[rF+*&FfgwF+F_g lF+F`hwFavQ,EIS~A052696F$"$X',$*(,&FboF+F*F4F+Ff]lF;FieoF;F;7$-F>6,Fg^p7%FB<#/F B-FV6$Fd^lFdfoFis77F5F4FhpF:FabnFbbnFcbnFdbnFebnFfbnFgbn"%\?"%(4%"%$>)"&&Q;"&pF $"&Pb'"'t58"'X@E"'*GC&"(x&[5F\[x<%,(FeqFboFfuF+F+F+/F\pF4Fibn,&F+F+F`hoF+F`hoF^ dpQ,EIS~A000051F$"$i$-F>6,F`el7%FBFc[xFgn77F5F4F]blF\\l"$3%"%gR"&+o%"'g,l")SAO5 "*Su:'="+++_>P",+K_*y")".+_bpC'>"/+WT")z,^"1+?J,jTG9"2+?*\O"=^G%"4+g&)*)3)o@r8" 5+!3q#GE=5iY"7++cm!)3b-Ny;"8+![/7Y#[!)=xP'":++GTm$y-*3&3^DF\[x<%Fd\xFibn,(*&,(F \qF5F]qF2FdpF+F+FeqF+F+*&,&F1F+F]qF^vF+FfuF+F+FjzF+*&Fe\xF+F_glF+F`hoFavQ,EIS~A 052561F$"$.&*&F*F+-Fc[l6#-F^cl6$*()F*&F[epFjoF+-Fc[l6#-F^cl6$*()F\_x&F[epF`pF+- Fc[l6#-F^cl6$*&)Fc_xFjdpF+FjdpF;FbepF+Fd_xF;/Fd_xFcepF+F]_xF;/F]_xFcepF+7#-F>6, Q&TreesF$7%FB<&/%#T1G-FL6$FN-FF6#%#T2G/%#T3GFN/Fi`x-FL6$FN-FF6#F[ax/FB-FL6$FN-F F6#Fd`xFis77FinF+F+F5FdpFiqFcdmFhjpFejrFabqFj[v"$>$"$l&"%,5"%\<"%ZI"%k_"%a!*"&n a""&?j#"&KX%Ff^xF^rF^rF^rQ=Unlabelled~trees~of~height~3F$Q,EIS~A001383F$F\]s*&, (F*F;-F_jl6#*&Ff]lF+FieoF;F;*&FebxF+F*F+F+F+Ff]lF;7#-F>6,F^hl7%FB<%/Fbs-FIFchlF [gmFcdwFgn77FinF5FhpFf`mF[[q"$k)"%!G)"&!['*"(?bK"")+k'4#"*S]&[P"++/2ou"-+wHQR;" .+k7H(HR"0+w0gW5-""1+%=@uyy&G"2+!on\i$>d)"4+S9Q3oEFu#"5+gXvq$=PZK*"7+!oZUb"[H!o N$"9++/">*>jSWcv7Fcbx<(,(*&,&F`qF5F]qFboF+FeqF+F+*&,(F\qF^vF]qF5FhpF+F+FfuF+F+* &,&F]qF+FboF+F+FjzF+F+F_uF]am/F_pFhp/FioFf`m/FbpF[[qF^r*&F]qF;F`hoF+FavQ,EIS~A0 52850F$"$=)*(F^[xF+Ff]lFboFieoF;7#-F>6,Ffan7%FB<#/FB-FL6$-FV6$FdfoFd^lFd^lFis77 F5FhpF]blF\_lF_blFablFd\r"$j#"$?&"%L5"%e?"%2T"%/#)"&(R;"&#yK"&^b'"')3J""'h@E"'1 V_"(&f[5"(sr4#F_ex<%Fibn,*FeqF5FfuF;F+F+F]qF;/F\pFhp,&F]qF+)F5FhuF+,$F`hoF5FavQ ,EIS~A052944F$Fiax-Fc[l6#-F^cl6$,$*&,&F;F+*$,&F+F+FaepF[yFfrF+F+FjdpF;F\vFbep7# -F>6,F^hl7%FB<%/FB-FFFa]w/F_s-Fe^lFa]w/FbsF_hvFis77F+F+F5FdpF]blF^blFjdmF]`l"$e ("%/D"%$[)"&.#H"'I?5"'U/O"(EfG""(-^i%")-Vv;")Iu1h"*vP!QA"*$**)=C)"+6,F^hl7%FB<$/FB-FL6 'FNF_sF_sF_sF_s/F_s-FF6#FBFgn77FinF+FiqFe^m"%'4%"'++;"(CE'z"*/.*=[",o$Q(fV$".cu !*46#G"0++++S9i#"2w"=%Q4O(>F"4_6;57)HS;J"6'*G>]i[1q(3R"8;C8(=.Jx'HlK&":+++++++' 4kTOy"=CU7**[F!Q&GR+%zB""?wt>GmJ)4q.pF#)**3#"A7T29Z)=k2U5WYn`v$"Cwp[zS%*3>BM'yj -xb:("F+++++++++?(e&e2)=:T9FcixF^rF^r,$*.F[vF\vF5Ffr-Fc[lF]pFfr-Fc[l6#F;FfrF]qF ]v-Fc[l6#*&,&F+F+FejlF+F+F]qF+F+FbrFavQ,EIS~A052764F$"$@(,$*&Ff]lF+,*F+F+F*FboF 9FboF[alF5F;F;7#-F>6,Ffan7%FB<#/FB-Fe^l6#-FL6$-FV6$-FL6$Fc`mFNFd^lFNFis77F+F+F5 F2FjqF^blFihpFd\l"$#H"$_'"%[9"%;K"%_r"&'*e""&G`$"&G&y"'WX<"'_zQ"'/B')"(Sm">"('4 gUF`\y<'FdoF^uFdflFdz,*FeqF5FfuFboF[rFboFjblF+-F^cl6$,$*&,*F[fuF+FcclFiflFdclFe dmFfclFebvF+FhclF+#F;"$F#/Fccl-F-6#,*F+F+F0FboFj]oFboF_dlF5,$*(,&F;F+-F-6$F[_y$ "+SY/*\%FgdlF+F+,(F__yF5*$F__yF4F2*$F__yFdpF7F;),(F5F+*$F__yF5F5Fd_yFboF]qF+F;F avQ,EIS~A052971F$"%V5,$*(FafmF+F*F+Ff]lF;F;7#-F>6,F^hl7%FB<%Fcdw/FbsF_gm/FBFa\n Fgn77FinFinF5F:F]hrFebl"%W;"&[B""'WX5"'ci)*")gdE5"*S]"p6"+!3ElW"",SuvB$>"-gliQs F".+/r%)>D%"/+c-3mUp"1+7p+G[-7"2+cUI1I@?#"3+/">6tF;D%"4+!oFNOHIJ')F[`y<&FdflF_u F[p,(*&,(F`qF+F]qF5F\qF4F+FeqF+F+*&,(F]qF\cuF\qFboFboF+F+FfuF+F+*&FhuF+FjzF+F+F ^rF^[nFavQ,EIS~A052881F$"$`)*&Ff]lF+,*F;F+F*F5F9F5FjxF;F;7#-F>6,Fav7%FB<#/FB-Fe ^l6#-FV6$-FL6&FNFNFc`mFd^lFNFis77F+F+F+F4Fe]lFddoF`crFfdmFb_l"$0#"$4%Fgdu"%R;"% xK"%`l"&2J""&:i#"&HC&"'d[5"':(4#"'J%>%F]by<&FdoF^uFgbl,*FeqF5FfuF;FjzF5F[rF;-F^ cl6$,$*&,(F+F+FcclFcdmFfclF2F+FhclF+#F+F`bl/Fccl-F-6#,*F;F+F0F5Fj]oF5FjemF;,$F` ho#F5FhpFavQ,EIS~A007909F$Fd]v,**&F*F5FfglF4F+*&F*F5FfglF5F^v*&FfglF+F*F5F4FjxF ;7#-F>6,F^hl7%FB<$/F_sFhhl/FB-FL6'FNFNF_sF_sF_sFgn77FinFinFinFinFinFcen"%!3""%+ jFb`s"'K+8"'S;_"(+l*>"(g$*Q(")OpnE")#R'[%*"*+vkH$"++o6O6"+K[;wQ",'\N@68",+pXJS% "-+G%4#p9Ffdy<)F_uF[pF^pFapFfav/FfpFcen,**&,*F\qFbqF]q!#mF`qF1FbqF+F+FeqF+F+*&, *F`qFgimF\qF]hrF]qFgimFgfyF+F+FfuF+F+*&,(F`qF1F\qFhzF]qFd]mF+FjzF+F+*&,&F`qF+F] qF;F+F[rF+F+F^r,$*0F[vF\vF5Ffr-Fc[l6##FboF4F4-Fc[lF`pF+F]qFb^n-Fc[l6#*&,&F+F+-F _jlFjoF+F+F]qF+F+)F]qFfjlF+#F+Fd]mFavQ,EIS~A052777F$"$M(*&F[[mF+F_^pF;7#-F>6,F` el7%FB<#/FB-Fe^l6#-FL6$FNFd[mFgn77F+F+F]bl"$]""%+I"&+]("(++D#")++vy"++++]J"-+++ ]<9".++++v3("0++++D")*Q"2+++++v)QB"4+++++vo-_""6+++++]7)=k5"7+++++]P4T")z"9++++ +++](G^Q'";+++++++]P%ftU&"=++++++++v$\BY)["?++++++++D1>BRSY"A+++++++++D1>BRSYF] hy<%FdoF^u,&*&,&F]qF\cuF\cuF+F+FeqF+F+FfuF+*&)FhpF^hoF+F_glF+,$Ficp#F+FhpFavQ,E IS~A034325F$Ffco,$*&FfglF;Ff]lF;F;7#-F>6,Q-derangementsF$7%FB<#/FB-FF6#-FI6$FN1 F5FSFgn77F+FinF+F5F:F]hr"$l#Fifr"&L[""''\L""(h\L"")qXo9"*T[@w""+KHz!H#",\55r?$" -Md^m5[".XJY,&*)Fdjy<%FdoF[p,(FbhnF +FduF+FjzF+F^r*$Fg[yF;QgqCycles~>=2,~permutations~of~n~element~with~no~fixed~po ints;~derangements~(subfactorial~or~rencontre~numbers)F$Q,EIS~A000166F$Fc\n,$*( ,(F;F+F*F+FjxF+F+FieoF;Ff]lF;F;7#-F>6,Fav7%FB<%/Fbs-Fe^l6$FN2FinFSF`hx/FB-FV6$F bsF_sFis77F+F5F4FhpF:FabnFbbnFcbnFdbnFebnFfbnFgbnFh[xFi[xFj[xF[\xF\\xF]\xF^\xF_ \xF`\xFf\z<&Fc\xFdoFdhmF]am,&F+F+F]hoF+F_hoFavFf\xF[[q,$*(F*F+,&F7F+F*F4F+Ff]lF boF;7#-F>6,Fj]l7%FB<#/FB-FV6$-FL6%Fb_pFd^lFd^l-FL6$F__pFd^lFis77FinFiqFddoFd]mF cdoFh`mFbbnF`rF^_lFeclFejrF[\sFhdmFf^sF]`nFg\qF]^qFfinFj`qFabqFb_lFh]z<&F_u,(Fe qF+F^vF+F]qF\cu/F\pFiq/F_pFddo,&F4F+F]qFhp,$F]qFhpQ)~5~n~+~3F$Q,EIS~A016885F$"$ V**&FieoF+F_^pF;7#-F>6,Ffan7%FB<#/FB-Fe^l6#-FL6$F__pFdfoFis77F+F4FcdmFj[qFc_v"% v="%v$*"&vo%"'vVB"(v=<""(v$fe")voHH"*vV[Y""*v=UK("+v$4@m$",voa5$=",vVt_:*"-v=nj xX".v$f$=))G#"/voz"4W9""/vV)*e/AdFa_z<%FdoFd\x,&FeqFhpFfuF;,$F`jyF4,$Ficp#F4Fhp FavQ,EIS~A005053F$"$$))*&FdcmF;,&F;F+F\[oF+F;7#-F>6,Fhcm7%FB<#/FB-FL6$F^dm-Fe^l 6#-FLFc_pFis77F+FinFinF+FinF+F+FinF+F+F+F+F+F+F+F5F+F+F5F+F5Fdaz<)FdoF[pF^pFapF i\l/FfpF+,4FeqF+FfuF5FjzF4F[rF4FjblF4Fe\oF5Fb`oF+Fa`lF+F]qF;,*F]q#F+Fcdm#FdpFcd mF+-F^cl6$,$*&,*FcclF;FdclF4FfclF+F5F+F+FhclF+#F;F`cr/Fccl-F-6#,,F_dlF+Fj]oF+Fj emF+F0F+F+F+F+-F^cl6$,$*&,&F+F+FcclF5F+FhclF+Fd_wFfemF+,$F]qFfbzQWnumber~of~way s~to~make~n~cents~with~coins~of~3~5~centsF$Q,EIS~A008676F$"$<#,&*&FafmF+FfglF+F +FafmF;7#-F>6,F^hl7%FB<%Fb`yFc`yF_eyFgn77FinFinF5F2Fd]mFcbn"$&H"%_;"&K5""&``)"' .#\("(%QZt")'\k&z"*T5aU*",F$>877"-KP@X"o"".3!4mF-D"/DAk+7wR"0:*Q[Bb=n"2O8\g]iG? ""3O8z>Vc^uAF[dz<)FdflF_uF[pFdz,.*&,(F\qF^vF]qFboF`qF;F+FeqF+F+*&,*F]qF\_lF`qF5 F\qFgimF]blF+F+FfuF+F+*&,*F`qF;F\qFhzF]qF]fuFbqF+F+FjzF+F+*&,(F\qFdpF]qF^blFj[o F+F+F[rF+F+*&,&!#6,Ffan7%FB<#/ FB-FL6$Fdfo-FV6$FNFd^lFis77F+FdpF:F\_lF]`oF`_l"$f"FgaxF]dr"%z7"%fD"%>^"&R-""&z/ #"&f4%"&>>)"'RQ;"'zwK"'f`l"(>2J""(R9i#Fefz<&FdoFhbw/F_pF:,(FeqFboFfuF+F;F+,&F;F +F]hoFhp,$F`ho#FhpF5FavQ,EIS~A052549F$"$'[-F>6,F`el7%FB<#/FB-FL6$F`gzFdfoFgn77F +FdpFd]mF[[q"$O*"%![*"'![9""(gxg"")![kd#"*?N7k%"++#*4'G*"-+#*4MV?".+CQ(\/\"0+K' *eJ_F""1+3EHctqN"3+?>_lPBr5"4+?.gC(*ozU$"6+S=p5v+8b;""7+?N'>7%HK&e>%"9+!3-(QwiV aU%f"":++'\)Hc7u?/xP'Fefz<&FdoFhbw/F_pFd]mF^^x*&FchzF+F_glF+FdhzFavQ,EIS~A05261 7F$"$i&,$Fc]mFgim7#-F>6,Q0Multiples~of~11F$7%FB<#/FB-FL6%-FV6-FNFNFNFNFNFNFNFNF NFNFNFd^lFd^lFis77FinFgimFg`mFbbnF]hrF`]lF]^sFjdmFfinFfbqFheqFjhqFd\lF^_rFd`nF\ gp"$w"Fi_lFh^m"$4#F^jnFfjz<%F_u/F\pFgim,&FeqF+F]qFd[w,$F]qFgimFh[[lFjjzQ,EIS~A0 08593F$"$B$,&FfrF+*$,(F+F+F*F[yFjxF[yFfrF\v7$-F>6,F[s7%FB<%/Fbs-FL6$FBFB/FB-FV6 %F_sFbsFN/F_sFdipFis77FinF+F5FdpFjqF[amFe^m"$W&"%G@"%W&)"&3]$"'#zX""''H:'"(#zDE ");;J6")_V7\"*G&Q[@"*W,NX*"+)G7C=%",/ZA$f=",%=L^,$)F[\[l<&F^uFdflF_u,(*&,&F]qFd pF[yF+F+FeqF+F+*&,&F5F+F]qFdpF+FfuF+F+*&F^buF+FjzF+F+F^r,$**F[vF\v,(FboF+*$F5Ff rF5*$,&Fc^[lFfrF\vF+F5FiqFfrF]qF]v)Fe^[lFfjlF+F`[lFavQ,EIS~A025227F$"$b'-F>6,Ff v7%FB<%/FB-FV6%F_sFNFbs/F_sFd\[l/FbsFdipFgn77FinF+FdpFcelF]_m"%+["'!o.""(ge"0+OZoxs%H"2+OZ+9'3N;"3+#z0f`t7')*"5+g`4]&flQU'"7+ S'[?">"Q@]\%"9+S#eCV&p8;\iL";+Sm&=$)[?fkOxn#"=+!Gd=Uc0Ugou?F [\[l<&,(*&,&F\qF[yFdpF+F+FeqF+F+*&,&F]qF[yFboF+F+FfuF+F+FjzF+F^uF_uFa[qF^rF`^[l FavQ,EIS~A052727F$"$$o*.Ff]lF;F_hpF;Fj`lF;F]_oF;,&F;F+*$F*FcuF+F;,&F;F+*$F*F[tF +F;7#-F>6,Fhcm7%FB<#/FB-FL6(Fd^lFbipFhalFg_o-Fe^l6#-FL62FNFNFNFNFNFNFNFNFNFNFNF NFNFNFNFN-Fe^l6#-FL6BFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFN FNFNFNFis77F+F+F5F5FdpFdpF2F2F]blF]blFb\lFb\lFf`mFf`mF^blF^blF_blF_blF\`nF\`nFi hpF[a[l"F]\t"'+k6Fa\tFch [lFe\tFbh[lFi\tF`h[lF]]tF_h[lFa]tFah[lFe]tF^h[lFi]tF]h[lF^^tFag[lFb^tF`g[lFf^tF [h[lFj^tF\h[lF]cpFjg[lFjbpFig[lFe`oFhg[lFa_tFbg[l,LF]q#"(t[b$"(?V'yFcq#Fc\n"')G C&F\q#"&rO""'W@EF`q#"%d<"'KkyF[as#F+"(g@$RF]au#Fc\nF\i[lFaau#"&@t#Fih[lFd`u#F^g pF_i[lF``u#"'.#="Fih[l#"'tbSFih[lF+-F^cl6$,$*,FcclF+,&F+F+FcclF+F+F[huF+FhuF+F\ [lF+#F;"&oF$/Fccl-F-6#,&F_dlF+F+F+F+-F^cl6$,$*&,>FdclF+Fa]oF+FdaoF5FeaoF5FdduF4 FjduF4FbduF4F\euF4FhduF4FafuF4FbfuF5FcfuF5FdfuF+F;F+F+FhclF+#F;Fgao/Fccl-F-6#,& *$F0FcuF+F+F+F+-F^cl6$,$*,F^j[lF+FccuF+FhuF+F\[lF+F]rF+#F+"';KR/Fccl-F-6#,&Fjem F+F+F+F+-F^cl6$,$*&,*FcclFhinFfclFcenFdcl"$P)!$w(F+F+FhclF+#F+F`j[lFaj[lF+-F^cl 6$,$*(,*FdclFdpFfclFdpFccl!#f!#hF+F+F]buF+FhuF+Fa\\lFaj[lF+-F^cl6$,$*&,&Fccl"%r 7!%36F+F+FhclF+#Fhp"'s58Ff[\lF+-F^cl6$,$*(,&Fg_lF+FcclFf_lF+F]buF+FhuF+#F\cuFa] \lFf[\lF+Fh`u#F+"(kGd"-F^cl6$,$**,&FcclFi`mF[fuF+F+F[huF+FhuF+F\[lF+#F;Fa]\lFf[ \lF+-F^cl6$,$*(,.FfclF+FdclF+Fa]oF+FdaoF+FcclF+F+F+F+F]buF+FhuF+#F;"%[?/Fccl-F- 6#,&*$F0FiqF+F+F+F+-F^cl6$,$*&,2Ffcl!#IFdcl!#DFccl!#JFdaoFiqFdduF[`nFa]oFjflFea oFiqFbqF+F+FhclF+#F+Ff^\lFg^\lF+,$F[asF`i[lQ[onumber~of~ways~to~make~n~cents~wi th~coins~of~1~2~4~8~16~32~centsF$Q,EIS~A008645F$"$X#,$*,FeazF;,&F;F+*$F*F]blF+F ;,&F;F+*$F*Ff`mF+F;,&F;F+*$F*F`blF+F;,&F;F+*$F*F_`nF+F;F;7#-F>6,Fhcm7%FB<#/FB-F L6'F^bz-Fe^l6#-FLFb`p-Fe^l6#-FL66FNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFN-Fe^l6 #-FL6TFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNF NFNFNFNFNFNFNFNFNFNFNFNFNFN-Fe^l6#-FL6`qFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNF NFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFN FNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNF NFNFis77F+FinFinFinFinF+FinFinFinFinF5FinFinFinFinF5FinFinFinFinFdpFi_\lF`q"$I(F]as!'+g pFaas!'++sFeas!'+?nFias!'+SiF]bs!'+!['Fabs!'+?bFebs!'+gdFibs!'++gF]cs!(+a9"Facs !(+S8"Fecs!(+w5"Fics!(+97"F]ds!(+E4"Fads!(C!f5Feds!(+k2"Fids!(g4-"F]es!(?0/"Fae s!'!3z*Fees!(S/+"Fies!'?V$*F]fs!'+q&*Fafs!'!)Q')Fefs!'Sw))Fifs!'g6"*F]gs!'w*R)F ags!'+g")Fegs!'+?zFigs!'+SuF]hs!'+!o(Fahs!'+u5Fehs!'w49Fihs!'+O7F]is!&+g'Fais!& +C*Feis!&+'yFiis!'+SQF]js!'?hLFajs!'C+OFejs!'!ol#Fijs!'S))GF][t!'+ICFa[t!'g&*>F e[t!'?4AFi[t!'S!z"F]\t!'![f"Fa\t!'+S]Fe\t!'++[Fi\t!'+gXF]]t!'+!3%Fa]t!'+?VFe]t! 'gBJFi]t!'+!G&F^^t!&+a$Fb^t!&+W%Ff^t!&+w#Fj^t!&w4#F]cp!&![:Fjbp!&S5"Fe`o!%gvFa_ t!&+Y&Fe_t!(?X="Fi_t!(g*)="F]`t!(C!z6Fa`t!(+C<"Fe`t!(+Y;"Fi`t!(+c:"F]at!(w**>"F `at!(S'*>"Fcat!(!))*>"Ffat!(?$)>"Fiat!(g"*>"F\bt!(++?"F_btF\g]lFbbtF\g]lFebtF\g ]lFhbtFec]lF[ctF\g]lF^ctF\g]lFactF\g]lFdctFdc]lFgctFhc]lFjctFic]lF]dtFfc]lF`dtF gc]lFcdtF`c]lFfdtFac]lFidtF_c]lF\etF^c]lF_etF]c]lFbetFjf]lFeetF[g]lFhetFbc]lF[f tFcc]lF^ft!(+q>"FaftFaf]lFdftFbf]lFgft!(SC>"Fjft!(!3&>"F]gtFef]lF`gtF\g]lFcgtFf f]lFfgtFdf]lFigtFcf]lF\htFhf]lF_htFif]lFbhtFgf]lFehtF\g]lFhhtF\g]lF[itF\g]lF^it F\g]lFaitF\g]lFditF\g]lFgitF\g]lFjitF\g]lF]jtF\g]lF`jtF\g]lFcjtF\g]lFfjtF_g]lFi jtF]g]lF\[uF^g]lF`[uF^d]lFc[uF]d]lFf[uFjc]lFi[uF[d]lF\\uF\d]lF_\uFhb]lFb\uFib]l Fe\uFgb]lFh\uFeb]lF[]uFfb]lF^]uF`d]lFa]uFad]lFd]uF_d]lFg]uFde]lF[^uFce]lF^^uFbe ]lFa^uFhe]lFd^uFjb]lFg^uF[c]lFj^uF\c]l-Ffo6#,&F]qF+F`j\lF+Fie]l-Ffo6#,&F]qF+F\j \lF+F[f]l-Ffo6#,&F]qF+F\[qF+F_e]l-Ffo6#,&F]qF+F_h\lF+F]e]l-Ffo6#,&F]qF+Fdj\lF+F je]l-Ffo6#,&F]qF+F\f[lF+F`f]l-Ffo6#,&F]qF+F\gpF+Fed]l-Ffo6#,&F]qF+FadvF+Fgd]l-F fo6#,&F]qF+Fg_lF+Ffd]l-Ffo6#,&F]qF+Ff^mF+Fbd]l-Ffo6#,&F]qF+F\_]lF+Fdd]l-Ffo6#,& F]qF+Fh[]lF+Fbb]l-Ffo6#,&F]qF+F\\]lF+F_f]l-Ffo6#,&F]qF+Fe`nF+F^f]l-Ffo6#,&F]qF+ Fei\lF+F]f]l-Ffo6#,&F]qF+Fb^wF+F\f]l-Ffo6#,&F]qF+Fg^mF+Fjfl-Ffo6#,&F]qF+FjbvF+F ^b]l-Ffo6#,&F]qF+F[[]lF+Fabv-Ffo6#,&F]qF+Fj[vF+F`b]l-Ffo6#,&F]qF+Fc[[lF+Fab]l-F fo6#,&F]qF+Fh_lF+F_b]l-Ffo6#,&F]qF+Fg^nF+Ffe]l-Ffo6#,&F]qF+Fd`nF+F[e]l-Ffo6#,&F ]qF+Ff\vF+F\e]l-Ffo6#,&F]qF+F^]]lF+Fge]l-Ffo6#,&F]qF+Ff_lF+Fid]l-Ffo6#,&F]qF+Fi hyF+Fjd]l-Ffo6#,&F]qF+Fh]]lF+Fhd]l-Ffo6#,&F]qF+FiinF+Fee]l-Ffo6#,&F]qF+FjinF+Fc d]l-Ffo6#,&F]qF+FcgzF+Fae]l-Ffo6#,&F]qF+Fdg\lF+F`e]l-Ffo6#,&F]qF+Fhg\lF+F^e]l,: -F^cl6$,$*(FcclF+,2Fa`lF+F\euF;FdaoFa`lFdfuF;F`^vF1Fj\vF1Fe]vF4Fd^vF4F+FhclF+#F +"%+5/Fccl-F-6#,,*$F0F[amF+*$F0Fi`mF;*$F0Ff`mF+*$F0F]blF;F+F+F+-F^cl6$,$*,FjbuF +FhuF+F\[lF+F]rF+F\clF+#F+"*+++?"/Fccl-F-6#,&F;F+Fi]oF+F+-F^cl6$,$**F_cnF+FhuF+ F\[lF+F]rF+#F4"(+++"F\_^lF+-F^cl6$,$*(F_cnF+FhuF+F\[lF+#F]fr"(+++%F\_^lF+-F^cl6 $,$*&F_cnF+FhuF+#"%(4'"'++SF\_^lF+-F^cl6$,$F_cn#")zwK8")+++!)F\_^lF+-F^cl6$,$Fc _^l#F+")+++7/Fccl-F-6#,&Fi]oF+F+F+F+-F^cl6$,$Fi_^l#Fdbl"(+++)F^a^lF+-F^cl6$,$F_ `^l#"&$Q9Ffa^lF^a^lF+-F^cl6$,$F_cn#"'FEi")+++;F^a^lF+-F^cl6$,$**FcclF5,6F`crF+F daoF`crF\euF`crFd^vF`crF^\vF:Fe]vF`crFj\vFj[oFh\vF:FdfuF`crF`^vFj[oF+F]buF+FhuF +#F+Fb`^l/Fccl-F-6#,.*$F0F`blF+Fb^^lF5Fc^^lF5Fd^^lF5Fe^^lF5F+F+F+-F^cl6$,$*(Fcc lF+,6"%qEF+Fdao"%XDFd^v"%]BF^\v"$Y)Fh\v"$,)Fj\v"%\OFdfuFcc^lF`^v"%aQFe]v"%DAF\e uFbc^lF+FhclF+Fgb^lFhb^lF+,$FcqFj^^lQ\onumber~of~ways~to~make~n~cents~with~coin s~of~5~10~20~50~100~centsF$Q,EIS~A001343F$"$&=,$*&F_hpFboFdcmF;F;7#-F>6,Fhcm7%F B<#/FB-FL6%FbipFbipF^dmFis77F+FinF5F+F4F5FhpF4Fe]lFhpF:Fe]lFjqF:FcdmFjqFd]mFcdm Fg`mFd]mF^blF^d^l<'FdoFdflF[pFi\l,2FeqFboFfuF1FjzF7F[rF1FjblFboF\qF+F]qFgimFi`m F+,.F\q#F+Fcel#FcbnFe^mF+F]q#Fe]lFcel-F^cl6$,$*&F^j[lF+FhclF+Fd_wFfemF+FaauFbrF ``u#Fe]lFcu,$F\qF\e^lQYnumber~of~ways~to~make~n~cents~with~coins~of~2~2~3~cents F$Q,EIS~A008731F$F`\v,$*&F]^pF+,(F+F+F*F\cuFjxF4F;F;7#-F>6,Ffan7%FB<#/FB-Fe^l6# -FV6$-FL6$F]_pFNFNFis77F+F5Fe]lFhjpFhin"$L&"%$H#"%m)*"&^C%"'dE="'Kfy"(*o"Q$")\1 b9")y"3E'"*V*)Qp#"+"=?"f6"+wSV()\",P)4)f9#",dpuOB*"-u_I/tR".*\&G6&46,F`el7%FB<#/FB-FL6 $Fd^l-Fe^l6#-FV6$FNFdipFgn77F+F5FiqFc\lF]_m"%+C"&gP#"'g@F"(g"[N")S=*=&"*+;)=%)" ,+or3]""-+W(>r"H".+)3D%)Rh"0+_v_l8R""1+Sa#*GsxL"2++%=thsX()"4+SMTOwpeS#"5++'H>4 -)R2q"7++s9/CGZLa@"8++%ez4j%*RsrpF`i^l<&FdoF[hoF]am,(*&,*F2F+F`qF+F\qF2F]qFgimF +FeqF+F+*&,&F]qFboF1F+F+FjzF+F+F[rF+*&,&F;F+-F^cl6$,$*&,&FdpF+FcclF4F+FhclF+Fbj y/Fccl-F-6#,(F;F+F0F+FjemF+F+F+F_glF+*(,&F;F+-F-6$F^\_l$"+())R.='FgdlF+F;,&Fa\_ lF;*$Fa\_lF5FboF;),&F+F+Fa\_lF+F]qF+FavQ,EIS~A052646F$"$#f*$-F^cl6$*(FddpF+Fjdp F;-F_jl6#*&Fg\wF+F[dwF;F+FbepF57#-F>6,F^hl7%FB<%F^cxFcdw/FBF`sFis77FinFinF+FdpF ]blFg`mF^_lF[emFhg\l"$%H"$^&F_i^l"%T>"%mO"%bp"&CK""&p_#"&i$["&vG*"'S'y""'`WMF[] _lF^rF^rF^rQ=pairs~of~Cycles~of~SequencesF$Q,EIS~A052821F$F_d[l,$**FieoF+F]^pF; F*F+Ff]lF;F;7#-F>6,Fav7%FB<&FcdwFc`y/F_\n-Fe^lFbgm/FbsF]\nFis77FinF+F5FhpFb\lFj [oF_iq"$l$"%%4""%"G$"%U)*"&D&H"&u&))"'@dE"'irz"(&["R#"(aW<(")hL_@")#3qX'"*X-r$> "*M28"eFf^_l<&F^uFdflF_u,(FeqF^vFfuF+F+F+,&FfrF+FgcpF_em,$FgcpF_emFavQ,EIS~A007 051F$Fg_l-F-6#,(F0Fbo*&F0F+-Fc[l6#*&F*F+F0F+F+F+F+F+7#-F>6,F^hl7%FB<%/F_s-FL6$F NFB/FBFahxF`gmFgn77F+F+FhpF\`n"$J'"&'e6"'ptE"(eFW("*"4LGC"+5w)44*"-`a74UQ"/i%)4 ,g4="0R#[#p<**R*"2'QO#\5h)Q`"4P$G\%*ozk"H$"6Q%[=bK3'*[*=#"8RP7a1g!y_!Hc"":')pI9 )37v#pG<>"";T"*o0aagX;Ebn'*"=1iPQ!*o"z/!R(fOJ)"?NpJ\Y5N6b;5`oavFd`_lF^rF^rF^rFa vQ,EIS~A052894F$"$q),$*$Ff]lFa`lF;7$-F>6,Q6Binomial~coefficientsF$7%FB<$Fi[n/FB -FL6)F_sF_sF_sF_sF_sF_sF_sFis77F+Fe]lFh`mF[emF^[q"$i%"$C*"%;<"%.I"%0]"%3!)"&wB" "&k&="&Kr#"&g(Q"&kU&"&8Y("'Z45"''fM""'+r<"'I-BFhb_l<$Fdo,2Feq!$?(F``sF+F[asFc\n FcqFh_lF`q"$N(F\q"%C;F]qFiimFhgnF+,$*.FhuF+F\[lF+F]rF+F\clF+Fg\oF+Fd`oF+#F+Fhgn ,$F``sF[e_lQOArrangements~of~n~unlabelled~objects~in~7~rowsF$Q,EIS~A000579F$"$c #-F>6,Q-ArrangementsF$F^c_lFcc_lFhb_lFdd_lFid_lF\e_lQMArrangements~of~n~labelle d~objects~in~7~rowsF$Q,EIS~A001730F$"$m#,*FfrF+*$,&F+F+F*F[yFfrF\vF*F]v*&F*F+Fh e_lFfrFfr7#-F>6,Ffv7%FB<%Fc`y/F_s-FV6$FNFbsFddwFgn77FinFinFinF2Fa^mFdeyFeel"'+O X")S](>""*!)GCj$",+;/aC""-+74njZ"/+S=xK6?"0+3!fSL#H*"2+cQB=iIm%"4++sm([`#e_#"6+ +kkOuek)o9"7++?Jx#>AWz7*"9++OB1#o(yPhOg";++78i/7O7a6,Q3Pairs~of~sequencesF$7%FB<$/FBFedw/F_s-Fe^lF`[zFgn77Fin FinFinFinFcelFhyFdelFeelFfel"(!Gx@")+;SD"*+WL>$"++W,6V",+!3-Fi"-+K?h*e*"/+gT#4# p:"0+Saoi*>F"1+SM$*Riz\"2++#fe0c.'*"4+?JTlg@j%>"5++)G+RTLf8%F_h_l<(F_uF[pF^pFfa vF]gl,&*&,(F\qF;F]qF+F5F+F+FeqF+F+*&,&F]qF+F^vF+F+FfuF+F+*&F[j_lF+F_glF+F]qQMPa irs~of~sequences~of~cardinality~at~least~2F$Q,EIS~A052520F$Fbco,$*(F*F+Ff]lF;-F c[l6#-F^cl6$,$*(FaepF+Fg\wF;FjdpF;F;FbepF+F;7#-F>6,Fav7%FB<%F^]z/FB-FLFd]z/F_sF _hxFis77FinF+F5FdpFe]lFjqF\_lFi`mFedmF__lFjjpFe_l"$&>"$s#"$t$"$3&"$%o"$:*"%77"% (f""%(3#F_j_lF^rF^rF^rFavQ,EIS~A000070F$Fd^m,$*&,&F+F+F*FhpF+Ff]lF^vF;7#-F>6,Fh gv7%FB<&F[hvF^hvFdhv/FB-FV6(F_sF_\nF_\nF_\nF_\nF_\nFis77F+FiqFc\nF[amFcbnFhaqF_ \rFc[[l"$D#"$!G"$T$F\]x"$"["$g&F[[x"$O("$L)F`iz"%X5"%g6"%"G"Fj[`l<%FdoFj^z,*Feq F;F\qF4F+F+F]qFdp,(F\qF4F+F+F]qFdp,$F\qF4Q1~5-gonal~numbersF$Q,EIS~A000567F$F`_ m,$FafmF57#-F>6,F^hl7%FB<%F[gmF^gmFb`yFgn77FinF5F5FdpFjqFeclFhyFaiwFg`vFdg[l"'g ds"(+wD(")+O$)z"*+K+e*Fff_l"-+CecV<".+gt[`h#"/+gxzb%=%"0+?>c[P6("2+gX6uu/G""3+S m<3?!HV#Ff]`l<%F_uF]am,&FduF+FjzF+F^r,$*$F]qF;F5FavQ,EIS~A052849F$"$<)*&FafmF5F *F57#-F>6,F^hl7%FB<$F^gm/FB-FL6&F_sF_sFNFNFgn77FinFinFinFinFcelFcen"$g'"%+U"&)o I"';SD"(SAN#")![3T#"*kg,r#"+!oNhJ$",gx&z(Q%"-+'4gIC'".+#RVK0&*"0SQ;7`?a""1+#zpv nbl#"2+CAd"\AQ["3+?>S)[#)yH*F]_`l<(F_uF[pF^pFfavF]gl,(*&,,FcqF+F`qF;F\qF1F]qFdp FiqF+F+FeqF+F+*&,*F]qFe]lF`qFboF\qF+F1F+F+FfuF+F+*&,&F\qF+F]qF;F+FjzF+F+F^r,$F] [nF5FavQ,EIS~A052754F$"$5(*.Ff]lF;Fj`lF;F`hpF;F[`\lF;FahpF;,&F;F+*$F*Fd]mF+F;7# -F>6,Fhcm7%FB<#/FB-FL6(Fd^lFhalFeipF[a\lFhip-Fe^l6#-FLF_^mFis77F+F+F+F+F5F5F4F4 FdpFdpF2F2F:F:FgimFgimFcdmFcdmFf`mFf`mF`crFaa`l@Fai s"'!3f#Feis"'?lBFiis"'+g=F]jsFje`lFajsFhe`lFejsFge`lFijsFee`lF][tFfe`lFa[t"'+GK Fe[t"'?"=$Fi[tF\f`lF]\tF]f`lFa\t"&![mFe\t"&?l)Fi\t"'!34"F]]t"'+'f"Fa]t"'!oL"Fe] tFie`lFi]t"&+#\F^^tFbf`lFb^tFaf`lFf^tF`f`lFj^tF_f`lF]cpF^f`lFjbpFcf`lFe`oFde`lF a_tF[f`l,HFaau#"&8o""'S%H)Fd`u#"$f$"'K)[#F]au#Fabn"'?ZTF``u#"(6v9"")+;W7Fh`u#F+ "(?$)[#-F^cl6$,$*(F^j[lF+F]buF+FhuF+#F;"$o(Ff[\lF+-F^cl6$,$*&,&FcclF`crFjflF+F+ FhclF+#F+Fgg`lFf[\lF+-F^cl6$,$*&,*FcclF+FdclF+FfclF5F;F+F+FhclF+#F+Fe^m/Fccl-F- 6#,(F_dlF+FjemF;F+F+F+-F^cl6$,$*&,2FcclFdpFdclF4FfclF4Fa]oF5FdduF+FeaoF+FdaoF5F ;F+F+FhclF+#F+Febl/Fccl-F-6#,,F[_\lF+F]boF+F_dlF+FjemF+F+F+F+-F^cl6$,$*(,*FfclF abqFdclF_aqFcclFddmF[`nF+F+F]buF+FhuF+#F+"&GL#FabuF+-F^cl6$,$*&,*Fccl!$J)FdclFg hvFfcl!$k*!$%>F+F+FhclF+#F;Fhi`lFabuF+-F^cl6$,$*&,*FdaoF5FeaoF+FhduF+F;F+F+Fhcl F+#F+Ff^m/Fccl-F-6#,(FeeuF+F]boF+F+F+F+-F^cl6$,$**,*FboF+FcclF+FdclF5FfclF;F+F[ huF+FhuF+F\[lF+Fgi`lFabuF+F]q#"',-YFag`lFcq#FabnFgf`lF\q#Fi_l"%;#*F`q#F_\r"'S#Q "F[as#F+"(+3A'#"((eZA"(+s9%F+,$F[asFh[alQ\onumber~of~ways~to~make~n~cents~with~ coins~of~1~4~6~10~12~18~centsF$Q,EIS~A008666F$F^`l*&F*FdpFfglF57#-F>6,F^hl7%FB< $/F_s-FFFf^l/FB-FL6(FNFNFNFNF_sF_sFgn77FinFinFinFinFcelFhyFaiw"%?n"&!)o#"&on*Fi fo"(gP,""(!GTI"(?fy)")w0gC")![#4n"*!G8*y""*Sq#zY"+g4C.7"+K/@[I"+!3E0i(F`\al<(F_ uF[pF^pFfavF]gl,&FhgoF+Fji_lF+F^r,$*0F[vF\vF5Ffr-Fc[l6#FboF5-Fc[lFcpF+F]q#Fe]lF 5-Fc[l6#*&,&F+F+FjdnF+F+F]qF+F+FigyF+#F+F[tFavQ,EIS~A052796F$"$`(,$**F]^pF+F[[m F;F*F+FieoF;F;7#-F>6,Fav7%FB<'Fj[n/FB-FL6$F_\nF_s/F_s-Fe^l6#%"FG/F_\n-Fe^l6$Fbs Fjhl/Fe_al-Fe^l6$F_\nFjhlFis77FinF+F4F]blF_blFb`n"$G&"%!3#"%c#)"&'*G$"'G88"'+[_ "(w")4#"(c1R)")G&eN$"*?fAM""*'H()o`"+;k^Z@"+G,+!f)",S%p)fV$"-;c@Ru8Fh^al<&F^uF_ uFdhm,(FeqFiqFfuF1FjzF+,&F^]mFbrF`hoF`[l,$F^]mFbrFavQ,EIS~A007582F$F`j\l,$*(F*F +,&F\cuF+F*F+F+Ff]lFboF;7#-F>6,Fj]l7%FB<#/FB-FV6$-FL6%Ff[mFd^lFd^lFg_nFis77FinF hpF:FddoFabnFc\nF`crFhjpFbbnF[`nFj[oFedmFcirFejrFgdmF[\oFcbnFablF]`nFjdmFd]qFaa al<&,(FeqF+F;F+F]qF[yF_uFahzF[gx,&F+F+F]qFdp,$F]qFdpQ)~4~n~+~1F$Q,EIS~A016813F$ "$Q*,$*(,(F*FboFjxF5F;F+F+FieoF;Ff]lF;F;7#-F>6,Ffan7%FB<#/FB-FL6$FdfoFg[oFis77F +FhpFgimFcdoFjhrFbblFabo"$$Q"$n("%N:"%rI"%Vh"&(G7"&vX#"&^"\"&.$)*"'2m>"':KR"'Jk y"(jGd""(Fd9$Ffbal<&FdoFbhzF[gx/F_pFgim,&F;F+F`hoF4,$F`hoF4FavQ,EIS~A052940F$"$ J*,$*(,&Faj_lF+F;F+F+F*F+Ff]lF;F;7#-F>6,F^hl7%FB<%FcdwFc`yF`gmFis77FinFinF+F4F2 FgimFd]mFhjpF]hrF]^sFhaqFf]r"$%>"$r#"$s$FehnFj`[l"$9*"%67"%'f""%'3#FfdalF^rF^rF ^rFavQ,EIS~A026905F$"$+)-F^cl6$*(FddpF+FjdpF;-F_jl6#*&Fg\wF+,(FaepF+F;F+)F*,$Fj dpF5F+F;F+Fbep7#-F>6,F^hl7%FB<%/F_s-FL6$FbsFNFj[n/FBF_cxFis77FinFinF+F+F5F5FdpF dpFe]lF:Fb\lFd]mFi`mF[amFhdmFj`qFi^rF^[q"$G$"$#\"$l(FhealF^rF^rF^rFavQ,EIS~A032 190F$"$k(*&Ff]lF;FieoF;7%-F>6,QDStirling~numbers~of~the~second~kindF$7%FB<#/FB- FLFf[xFis77F+F4Fe]lFcdmF]_lFhdmFd_l"$b#Fecr"%B5"%Z?"%&4%"%">)"&$Q;"&nF$"&Nb'"'r 58"'V@E"'(GC&"(v&[5"(^r4#F`gal<$FdoFbhz,&F;F+F]gxF+F^gxQWPartitions~of~a~set~of ~size~n~into~2~non-empty~subsetsF$Q,EIS~A000225F$"$X$-F>6,Q/Sums~of~powersF$Feg alFigalF`galFghalFhhalF^gxFavFjhalF]]v-F>6,F`el7%FBFfgalFgn77F+F4Fb\lFb^m"$W("% gv"&S9*"(+_G"")?Ng?"*SiAr$"++O:Gu"-+gHfM;".+c5-N#R"0+kw"Gh6,av7":++k_(['\$zG;-^F`ga l<%FdoFd\xF^^x*&FhhalF+F_glF+F^gxFavQ,EIS~A029767F$"$)\**Ff]lF;F_hpF;Fj`lF;F]_o F;7#-F>6,Fhcm7%FB<#/FB-FL6&Fd^lFbipFhalFg_oFis77F+F+F5F5FdpFdpF2F2F]blF]blFb\lF b\lFf`mFf`mF^blF^blFddmFddmF]hrF]hrFb[sFhjal<.FdoF^uFdflFac[lFbc[lFcc[lFdc[lFec [lFfc[lFgc[lFhc[l,BFeqF2FfuFd]mFjzFi`mF[rFc\lFjblFeclFe\oFeclFb`oFeclFe`oFeclFj bpFc\lF]cpFi`mFj^tFd]mFf^tF2F]q!$1&!%%=#F+F`qF;F\q!#R,6F`q#F+Fh[mF\q#FcdmF_e_lF ]q#FcdzFgg`l#Fh_lF_e_lF+-F^cl6$,$*&,&F;F+FdclF+F+FhclF+Fe^alFaj[lF+-F^cl6$,$Feg `lFjj[lFf[\lF+-F^cl6$,$*&,&FcclFe]lF1F+F+FhclF+FfaoFf[\lF+Fd`u#F+F_e_lF``u#Fcir F_e_lFaauFi[bl,$F`qFh[blQennumber~of~ways~to~make~n~cents~with~coins~of~1~2~4~8 ~centsF$Q,EIS~A008643F$"$L#,**&F*F4FfglF4F+*&F*F4FfglF5F^v*&F*F4FfglF+F4F9F;7#- F>6,F^hl7%FB<$F_ey/FB-FL6(FNFNFNF_sF_sF_sFgn77FinFinFinFinFinFinFhgnFdial"&+/&F `j^l"(?.I""(S!Qd")+!eR#")!ohg*"*/rZt$"+!)eH<9"+++Ou_",+c)RJ>",wp'4xp"-CWdI"\#"- +!Q"H1))F_]bl<*F_uF[pF^pFapFfavFepFgav,*FefyF+*&,*F\qFbbnF]q!#W!$K"F+F`qFgimF+F fuF+F+*&,(F`qF1F]qFc\lFbqF+F+FjzF+F+*&,(F`qF+F\qF^vF]qF5F+F[rF+F+F^r,$*0F[vF\vF 5FfrFh[yF4-Fc[lFjoF+F]qFehzFdgyF+FigyF+#F+F\\lFavQ,EIS~A052784F$"$T(*&Ff]lF;Fdc mF;7$-F>6,Fhcm7%FB<#/FB-FL6$Fd^lF^dmFis77F+F+F+F5F5F5F4F4F4FdpFdpFdpFhpFhpFhpF2 F2F2Fe]lFe]lFe]lFh_bl<%FdoF^u,,FeqF+FfuF+FjzF+F]qF;F^vF+,(F]qFj_w#F5F4F+-F^cl6$ ,$*&,&F;F+FcclF+F+FhclF+Fd_wFfemF+,$F]qFj_wQWnumber~of~ways~to~make~n~cents~wit h~coins~of~1~3~centsF$Q,EIS~A008620F$F^[q-F>6,F`el7%FB<#/FB-FL6$F^dmFd^lFgn77F+ F+F5FjqFeclFhyFdelF]g[l"'g47"(?:X"")+_^9"*+smf""++!3]R#Fjen"-+gX"*eV".+!3i/Yy"0 +!G$Rn`D""1+gdoX7M@"2+g4Sfh;[%"3+S#='Gq::&)"5++[Os09..6,Fav7%FB<#/FB-Fe^l6#-FV6%FNF^bnFc`mFis77 F+F4FgimF]`oFe_lFb]p"%j<"%zi"&jB#"&Z'z"'nOG"(&H55"(>#)f$")Z_"G"")z@kX"*Jqbi""*^ a&*y&"+:/)>1#"+Z@&QM(",rs^bh#",2hDaJ*Fibbl<%FdoFd\x,(FeqF5FfuF4FjzF;-F^cl6$,$*& ,&F4F+FcclFdpF+FhclF+#F+Fabn/Fccl-F-6#,(F;F+F0F4FjemF5,$*&,(#F:FdpF+*$FabnFfr#F ^vFdp*$,&FgeblF+FfeblF`[lF5F[yF;)FieblFfjlF+F;FavQ,EIS~A007482F$"$U%,$*(F^[xF+F h\zF;Ff]lF;F;7$-F>6,Fav7%FB<#/FB-FV6$-Fe^l6#-FL6%FNFNFd^lFd^lFis77F5F+F5F5F4Fdp F2F:Fb\lFg`mFddmFb[sFb^mFc`nFj^mFihv"$6'"$))*"%)f""%&e#"%#=%F]fbl<%F^uFibn,*Feq F;FfuF;FjzF+F+F+,&F+F+-F^cl6$,$*&,&F;F+FcclF4F+FhclF+FbjyF[\_lF+,$**,&FboF+Fa\_ lF4F+Fe\_lF;F`\_lF;Fg\_lF+F;FavQ,EIS~A001611F$"$-%-F>6,F`el7%FB<#/FB-FV6$Fd^lFg fblFgn77F5F+FdpFjqFa^mFbjvFcjvFfgm"'![k&"(gL)z"*+!3q7"++3MNA",+S95J%"-+g,=H!*"/ +39?(*R?"0+S564I%\"2+!o:iCQy7"3+![)e*y">9N"5+SM`<=$*4B5"6++sIob%e_WJ"8++[3Z>)>' Ru,"F]fbl<&F^uFibnFa[qF_[_l*&FcgblF+F_glF+FigblFavQ,EIS~A052661F$"$3'*&Ff]lF+,( F;F+F*F4FjxF+F;7#-F>6,Ffan7%FB<#/FB-Fe^l6#-FL6$Fd^l-FV6%FNFNFdipFis77F+F5Fe]lFc doFeinF\`v"$H)"%QF"%V!*"&n)H"&W')*"'*zD$"(Tg2""(ARb$")2yt6")VtwQ"*O)R!G""*^o)GU "+*Q+nR""+=!))Hh%",VWmN_"Fgibl<%Fdo,(FeqF+FfuF4FjzF;F]am-F^cl6$,$*&,&FcclFhpF+F +F+FhclF+F^h^l/Fccl-F-6#,(F;F+F0F4FjemF+*(,&#F\cuF5F+Ffh^lFfrF+,(Fd^nF+Ffh^lF]v *$,&F]vF+Ffh^lFfrF5FboF;)Fe\clFfjlF+FavQ,EIS~A052924F$"$4*,$*$,*F;F+F*F+FjxF5F9 F+F;F;7#-F>6,Fav7%FB<%/Fbs-FV6$%(EpsilonGFNFahl/F_s-FL6%FbsFbsFNFis77F+F+F4F2Fd doFh`mFihpFdbn"$x#Fefw"%y7"%XFFgfw"&kE""&,s#Fifw"'"\D""'U&p#F[gw"(CNC""(k4n#Fi\ cl<&FdoF^uFdhm,*FeqF+FfuF5FjzF+F[rF;-F^cl6$,$*&,(F1F+FcclFa`lFfclF5F+FhclF+F[hw /Fccl-F-6#,*F;F+F0F+FjemF5Fj]oF+,$*&,(*$-F-6$F\_cl$"+>BrbYFgdlF5F[yFa_clF;*$Fa_ clF4F^vF;),(F+F+Fa_clF5F`_clF+F]qF+F;FavQ,EIS~A002478F$"$7%**Ff]lF;FieoF;F]^pF; F[[mF;7#-F>6,Fdgal7%FB<#/FB-FL6&Fd^lFdfoF]_pFd[mFis77F+F]blFcbnFidw"%,<"%qx"&0T $"']d9"',:h"(ID`#")X6,Fhcm7%FB<#/FB-FL6%Fd^lF^bz-Fe^l6#-FL6;FNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNF NFNFNFNFNFNFNFNFNFis77F+F+F+F+F+F5F5F5F5F5F4F4F4F4F4FdpFdpFdpFdpFdpFhpF`bcl6,Q2Integer~partitionF $7%FB<#/FB-FF6$F[\n1FSFdpFis77F+F+F5F4FhpF2F:FgimFcdmFd]mFcdoF\`oFj`mF]`oFjhrF\ \lFgclFa^mF[emF_aqFc^mFhfcl<)FdoF^uFdflFhoF_`oF[ao,8FeqF2FfuFd]mFjzFi`mF[rF_blF jblFi`mFe\oFd]mFb`oF2F]q!$">!$/&F+F`qF;F\qFjfl,2F`qFch`lF\q#FhpFeclF]q#FcdmF[t# Fh_lF]_mF+-F^cl6$,$F_cnF`vFf[\lF+-F^cl6$,$Fi`bl#F+F\`oFfemF+FaauFe^alF``u#FhpF[ t,$F`qFch`lQBPartition~of~n~in~at~most~4~partsF$Q,EIS~A001400F$"$`$,$*&,&F;F+Fg e_lF+F+F*F;F\v7$-F>6,Fav7%FB<$Fahl/F_s-FL6$FBFNFgn77F+F+FdpFi`m"$O$Feen"&S]*"(g @;#")+wld"+gDKk<",+[K\4'".+[+\lM#"/+#>33Q'**"1++K_P0EY"3+!Gp64J:L#"5++'HB3.qxE" "6++'4CR.5"yR("8++%)>g9@>-8h%":++gtW5=PX@!eI"<++3%\WFQ'>3*y\@">++!)))4f*GJ6,F``x7%FB<$F`a_lFahlFis77F+F+F5FhpFb\lFc\lFd\ lFe\lFf\lFg\l"&'z;"&'ye"'7!3#"'+Hu"(SWn#"(X[p*")qwNN"*!zW'H""*+(QwZ"+!>jsw""+?/ 7klFgjcl<$Fdo,&*&F^^[lF+FeqF+F+FdawF+F^rF[h_lQ:Planar~trees~with~arity~1F$Q,EIS ~A000108F$Fee[l,$*(,(F*F;F9F+F;F+F+Ff]lF;F_hpF;F;7#-F>6,F`el7%FB<#/FB-FL6$Fd^l- FV6$FNFbipFgn77F+F5F2Fd]mFhaqFbjvF\g[lFiy"'?>CFjh_lF[i_l"*+wTz#"++G,KQ",+k;;)\" -+3i/Yy"/+?Jp!p<""0++)))*yA4#"1++'4Guob$"2+!3Iw5hUq"4+?:(\/h4Q8"5++o>")4C[>HFe\ dl<'Fdo/FioFd]mFdfnF]am,(*&,*F`qF;F\qF7F]qF^fzFgdlF+F+FeqF+F+FdawF+*&F\clF+FjzF +F+*&,(#Fe]lFdpF+F``uF`[lF]qFfrF+F_glF+Fi[lFavQ,EIS~A052687F$"$N',$*&Ff]lFbo,&F ;F+*$F*FgimF+F;F;7#-F>6,Fhcm7%FB<#/FB-FL6%Fd^lFd^l-Fe^l6#-FLFa[[lFis77F+F5F4Fdp FhpF2Fe]lFiqF:F]blFgimFddoFcdmFabnF\_lFc\nFcdoF`crF\`oFhjpF]_lFi^dl<-FdoF_`oFdh mFdc[lF]amF``o/FiapF:,>FeqFboFfuFboFjzFboF[rFboFjblFboFe\oFboFb`oFboFe`oFboFjbp FboF]cpFboFj^tFboF\qF+F]qFcdoFd\lF+/F_bpFiqFh`oF[ao,*F\q#F+Fg`m#FcuFgimF+F]q#Fd doFg`m-F^cl6$,$*&,4FfclFdpFdclFe]lFa]oF:FdaoF]blFeaoF]blFdduF:FjduFe]lFbduFdpF\ cuF+F+FhclF+#F+Fjhq/Fccl-F-6#,8Fe^^lF+*$F0F:F+F[_\lF+F\boF+F]boF+Fi]oF+F_dlF+Fj ]oF+FjemF+F0F+F+F+F+,$F\qF^`dlQZnumber~of~ways~to~make~n~cents~with~coins~of~1~ 1~11~centsF$Q,EIS~A008729F$F_eal*$-Fc[l6#,$*&F*F+Ff]lF;F;F57#-F>6,F^hl7%FB<%Fj[ nF^[`lFfh_lFgn77F+F5FiqF]hr"$/$"%7D"&kS#"'O:E"())Q<$")oR[U"*C%f6i"+3-&>%)*"-_$o #zy;".'p\NslI"/w&GN4^'f"13E")>6dJ7"2'\-I>YN)o#"3W*eq!=]j%='"5sIWd0]hB&\""6s)*zA #oa!*R*y$"8cUYm#Qbe))Q/5F_adl<%Fdo,(*&,&F\qF+F]qF+F+FeqF+F+*&,&F[yF+F]qFboF+Ffu F+F+FjzF+F]amF^r,$*,F[vF\v-Fc[l6#F\vF5F5F`[lF]qFgebl-Fc[l6#,$*&F5FfrF]qFfrF5F+F frFavQ,EIS~A052897F$"$t),$*&FjfclF+Ff]lF^vF;7#-F>6,Fhgv7%FB<&F[hvF^hvFdhv/FB-FV 6$F_sF_\nFis77F+FdpF:FcuF`crF_blFcirFgclFd]qF_`nFjhqFe^mF\j\lF\jnFf\`lF_e_l"$*G F__m"$h$"$+%"$T%F[ddl<%FdoFhbw,*FeqF;F\qF+F]qF5F+F+FjjmF\qQ1~1-gonal~numbersF$Q ,EIS~A000290F$"$Q$,$*(FjfclF;Ff]lF^vFifclF;F;7#-F>6,F_gcl7%FB<#/FB-FF6$F[\n1FSF 4Fis77F+F+F5F4FdpFhpFe]lFiqF]blFjqFb\lFcuF\_lFc\nFcelF\`oFi`mFbbnF[`nF[amF]hrF_ edl<&FdoF^uFdfl,0FeqFboFfuF[yFjzF[yF[rFboF\qF+F]qF:Ff`mF+,,F\q#F+Fjq#FjhrFa^mF+ F]qFfr-F^cl6$,$F_cnFd_wFfemF+F``uFbr,$F\qF^fdlQBPartition~of~n~in~at~most~3~par tsF$Q,EIS~A001399F$"$_$*(F*F5,(F*F;FjxF+FboF+F+Ff]lF;7#-F>6,F`el7%FB<#/FB-FL6%F NFN-FVFbfwFgn77FinFinFdpFd]mFeclFhyFaiwFg`vFdg[lF]^`lF^^`lF_^`lF`^`lFff_lFa^`lF b^`lFc^`lFd^`lFe^`lFf^`l"4++G`j,/e'[Fgfdl<(F_^dlF_uF[pFahnFa[q/FbpFecl,$F_glF5F 5FavQ,EIS~A052642F$"$)e*(Ff]lFboF_hpF;FdcmF;7#-F>6,Fhcm7%FB<#/FB-FL6&Fd^lFd^lFb ipF^dmFis77F+F5FdpFe]lFgimFcuFcdoF]_lFj[oFejrF__lF]^qFccqFdiqFg^nFh[]l"$/#Fb\vF himFaam"$e$Figdl<&FdoFa[qF]am,2FeqF2FfuFjqFjzFjqF[rF2F]q!#uF^b]lF+F`qF;F\q!#:,. F`q#F+F_blF\qF^e^lF]q#FgimFjq#FahqFe^mF+FbhclF+F``uF`v,$F`qFjhdlQennumber~of~wa ys~to~make~n~cents~with~coins~of~1~1~2~3~centsF$Q,EIS~A000601F$F\jn,$*(F*F+,&Fa `lF+F*F+F+Ff]lFboF;7#-F>6,Fj]l7%FB<#/FB-FV6$-FL6%Ff_pFd^lFd^lFg_nFis77FinFe]lFd doF\_lF`crF]_lF[`nF^_lFcirF`]lF[\oF__lF]`nF`_lFf^qFa_lFjjpFb_lF``nFc_lFjhqF`idl <&F_u,(FeqF+F;F+F]qF1/F\pFe]lF[_z,&F+F+F]qF2,$F]qF2Q)~6~n~+~1F$Q,EIS~A016921F$" $X**(,*FboF+F*FhpFjxF^vF9F+F+FieoF;Ff]lFbo7#-F>6,Ffan7%FB<#/FB-FV6$-Fe^l6#-FL6$ Fd^lFN-FL6$Fd^lFd^lFis77F5F4FhpFiqFddoFg`mF]`oFa^mFa]rFee_l"$B&"%O5"%h?"%5T"%2# )"&+k""&&yK"&ab'"'"4J""'k@E"'4V_Ffjdl<',(FeqF5FfuF;F]qF;Fd\xFibn/FioFiqFidx,(F+ F+F]hoF+F]qF+F_hoFavQ,EIS~A052968F$"%S5,(F*FfrFfrF+*$,(F+F+F*F1FjxF+FfrF\v7#-F> 6,Ffv7%FB<%/FBF`f_l/FbsFds/F_s-FL6$FBFbsFgn77FinF5FdpF_blF]`al"&+3""'!o$G"(SA5* "*g&e]M",!oF24:"-+CU))yu"/+C'=QA9%"1+O^."pb`#"3+/&*=Be$)*p""5+_$)=@H#='Q7"6+S)> /KQ_BZ(*"8+SuzWUE5k&Q#)":+gx6#*4>haENW("<+S5J'zD0/'*y+fr">+?Vu#)G?K9js%e-t"@++O :wxo%)ym64LY(yFh\el<&F_uFa[qF]am,(*&,&F\qF+F;F+F+FeqF+F+*&,&F^vF+F]qF1F+FfuF+F+ FjzF+F^r,$**F[vF\v,(Fd]mF+Fc^[lFhz*$,&F4F+Fc^[lFboF5FboFfrF]qF]v)F`_elFfjlF+F`[ lFavQ,EIS~A052716F$"$s',$*&F_hpF+,*F9F+FjxFboF*F;F+F+F;F;7#-F>6,Ffan7%FB<#/FB-F e^l6#-FV6$-FL6%FbipFNFNFNFis77F+F+F5F4F2F]blF\_lFbbnF[\oFc^m"$(>Fffdl"$Q'"%X6"% p?"%@P"%9n"&(37"&%z@"&a#R"&b2(Fd_el<&,*FeqF+FfuFboFjzF;F[rF+FdoF^uFdfl-F^cl6$,$ Fecz#F+Fe]l/Fccl-F-6#,*Fj]oF+FjemFboF0F;F+F+,$*(,&F;F+*$-F-6$Fgael$"+@8e\bFgdlF 5F+F+,(*$F\belF4F^vF\belF+F[belFdpF;),(F+F+F\belF5F[belF;F]qF+F;FavQ,EIS~A05298 3F$"%d5*&,&F;F+F\^pF+F+,(F;F+F\^pF+F*F+F;7#-F>6,F`al7%FB<#/FB-Fe^l6#-FL6$FN-Fe^ l6#-FLFf_nFis77F+F+F+F+F+F+F+F+F+F+F5F4FdpFhpF2Fe]lFiqF:F]blFjqFcdmFfbel<-,(Feq F+FjbpF+F]cpF;FdoF^uFgblFi\lFhblFcbz/F_bpF+/Fi`oF+/F\bpF+/FiapF+-F^cl6$,$*&,4Fc cl!(_r4#Fjdu")kOB@Fddu")sy)Q#Feao")cQ(o#Fdao"))3L-$Fa]o")CA,MFdcl")_PEQFfcl")@n /V")oV()=F+F+FhclF+#F+"*0x>/%/Fccl-F-6#,(F;F+F[adlF+F0F+*(,&F;F+*$-F-6$F`eel$"+ Kc+V#)FgdlF:F+F+,&FceelFdqFdeelF;F;),&F+F+*$FdeelFiqF+F]qF+FavQ,EIS~A005711F$"$ #Q*&F_hpF;FdcmF;7#-F>6,Fhcm7%FB<#/FB-FL6$FbipF^dmFis77F+FinF+F+F+F+F5F+F5F5F5F5 F4F5F4F4F4F4FdpF4FdpF^fel<&FdoF[pFgbl,.FeqF+FfuF5FjzF5F[rF+F]qF;F[yF+,*F]qF_em# FhpFjqF+FbczF+F``uF`[l,$F]qF_emQWnumber~of~ways~to~make~n~cents~with~coins~of~2 ~3~centsF$Q,EIS~A008615F$"$7#,$*$Ff]lF^vF;7$-F>6,F]c_l7%FB<$Fi[n/FB-FL6%F_sF_sF _sFis77F+F4F2F]blFcdmFc\nFh`mF_blFedmF`]lF]^sFg\qFa_lFadqFcenFb`nFf\vFei\lFfhvF ^[qFghvF`gel<$Fdo,*FeqFboF\qF+F5F+F]qF4,$*&FhuF+F\[lF+Ffr,$F\qFfrQOArrangements ~of~n~unlabelled~objects~in~3~rowsF$Q,EIS~A000217F$"$`#-F>6,Fbe_lFegelFjgelF`ge lF[helF]helF_helQMArrangements~of~n~labelled~objects~in~3~rowsF$Q,EIS~A001710F$ "$i#*$FfglFiq7#-F>6,Q,Powers~of~8F$7%FB<$Ff\al/FB-FL6*F_sF_sF_sF_sF_sF_sF_sF_sF gn77F+FiqFgclFhbblFdjxF`j[lF\i[l"(_r4#");sx;"*Gx@M""+C=ut5"+#fM**e)",OnZ>(o"-)) Q"ev\&"./6^Y!)R%"/K))3sV=N"0c1rw\Z"G"1[_o8)*z^A"2%)>[4&)R9!="3se&e2)=:T9"4wp%og /:#H:"Fhhel<$Fdo,&FfuF+FeqF7F^r,$*,F[vF\vF5FfrF]qF\v-Fc[l6#*&,&F+F+-F_jlFjapF+F +F]qF+F+FigyF+FfrQ\oArrangement~of~n~distinguishable~objects~in~8distinguishabl e~urnsF$Q,EIS~A001018F$F[^v,$*(,(FjxF;F9F+F;F+F+Ff]lF;FieoF;F;7#-F>6,Ffan7%FB<# /FB-FL6$-FV6$FdipFd^lFdfoFis77F+F4FiqFabnFddmFc_sF^_rFd_v"$v&"%^6"%.B"%2Y"%:#*" &J%="&jo$"&FP("'bu9"'6\H"'B)*e"(Z'z6"(&HfBF\[fl<'FdoFd\xF[hoFbhz/FioFabn,&F;F+F `hoFeebl,$F`hoFeeblFavQ,EIS~A052996F$"%u5,$*&Ff]lFboFeazF;F;7#-F>6,Fhcm7%FB<#/F B-FL6%Fd^lFd^lF^bzFis77F+F5F4FdpFhpFe]lF:FgimFddoFcdmFd]mFc\nFcelF\`oFi`mFj`mF` rFc\lF\`nF`blF`]lF]]fl<'FdoFdhmF]am,2FeqFboFfuFboFjzFboF[rFboFjblFboF\qF+F]qFgi mFi`mF+F``o,*F\q#F+F]blF+F+F]q#Fe]lF]bl-F^cl6$,$*&,(FfclF+FdclF+FboF+F+FhclF+#F +F`crF^czF+,$F\qF[^flQYnumber~of~ways~to~make~n~cents~with~coins~of~1~1~5~cents F$Q,EIS~A008732F$Fcco,$*$F\alF;F;7#-F>6,F`el7%FB<#/FB-Fe^l6#-FV6$FNFjalFgn77F+F +F5F2FeclFa_m"%!)GFiy"'SAGFjgn")+K!3&"*+#>%e(Fff_l"-+)[F"2+gd!3`]fY"4+!ontS'H)e6"5+++3A5v7TI"6++!3%4#G>^$R)Ff^fl<'FdoF^uFdflF eflFdz*&-F^cl6$,$*&,*F\`oF+FdclF_blFfclFeclFcclFgclF+FhclF+FiclF[dlF+F_glF+,$*& FhdlF;FidlF+F;FavQ,EIS~A052593F$"$Q&*$Ff]lFhz7#-F>6,F]c_l7%FB<$Fi[n/FB-FL6.F_sF _sF_sF_sF_sF_sF_sF_sF_sF_sF_sF_sFis77F+FjqFg\qF\\v"%l8"%oVFjc_l"&C=$"&#ev"'gz;" ';FN"'Kaq"(y?N""(Wh\#"(+uX%"(ghs(")&*y.8")!=u9#")!H(fM")+tia"):Bn%)Fj`fl<$Fdo,< FeqFien*$F]qFgimF;*$F]qF]blFgfy*$F]qF:!%D>F^bv!&qE$F]^t!'BuNF``s!(evj#F[as!)N&R L"Fcq!)Id*f%F`q!*w!e_5F\q!*wz"4:F]q!*SQa?"!)+o"*RF+,$*8FhuF+F\[lF+F]rF+F\clF+Fg \oF+Fd`oF+Fg`oF+F\cpF+F_cpF+F\_tF+Fh^tF+#F+Fien,$FfbflFecflQPArrangements~of~n~ unlabelled~objects~in~12~rowsF$Q,EIS~A001288F$Fd\v*&Ff]lF+,(F;F+F\[oF+F*F+F;7#- F>6,F`el7%FB<#/FB-Fe^l6#-FL6(FNFNFNFNFNFd^lFgn77F+FinFinFinFinFcenFhgnFeenFignF genF^^`lFhen"++k+;>Fjen"-+supI_"/+S%\Rh/""0+!o()o],B"1++W@9JN`Fahm"4+?jH1hsF;$" 5++w0!y#o'=F)Ficfl<(FdoF[pF^pFapFfav,(*&,.F\q!$D#F]q!$u#F^b]lF+F[asF;FcqFhhdlF` q!#&)F+FeqF+F+*&F[cuF+FjblF+F+Fe\oF+*&-F^cl6$,$*&,,FgclF+Fa]oFijpFdclF_`nFfclF] jqFccl!$*oF+FhclF+#F;"%"Q$/Fccl-F-6#,(F;F+Fi]oF+F0F+F+F_glF+*(,&F;F+-F-6$,(Fj]o F+FjemF+F;F+$"+imx[vFgdlF+F+,&*$FdfflFhpF\cuFdfflF;F;),&FdfflF+*$FdfflF5F+F]qF+ FavQ,EIS~A052627F$"$t&,$*(,&FboF+F*FhpF+F[[mF;Ff]lF;F;7$-F>6,Ffan7%FB<#/FB-FV6$ Fd[mFd^lFis77F5FhpFabnFcbnFebnFgbnFi[xF[\xF]\xF_\xFa\x"(0V>%")(o"-Xp!z([F".xxi6&*4"F`gfl<%Fibn,(FeqF[yFfuF+F4F +F[gx,&F^]mF+F+F+F^]mFavQ,EIS~A052539F$"$q%-F>6,F`el7%FB<#/FB-FV6$Fd^lFd[mFgn77 F5FhpFj`mFbe[l"%oh"'+I7"(S)\H")+/e#)"+S=XUE",+w#)H))yT"5+%y5vQWu,M#"7++gT?SiY5/9"8+gt#Hi4L)pi)*)":++w@OPQa[j16'"<+g`\' zdNK4xn*R%">++CQ@&R5%\!f]vVL"@++GHl95'HABZS+vEF`gfl<%,(*&,(F\qFdpF]qFjqFiqF+F+F eqF+F+*&,&F]qF\cuFgdlF+F+FfuF+F+FjzF+FibnF[gx*&FhhflF+F_glF+F^]mFavQ,EIS~A05269 5F$"$W',$*(FgfzF+Ff]lF;F]^pF;F;7#-F>6,Ffan7%FB<#/FB-FL6$-FVFc[elF]_pFis77F+FhpF cuFcirFiin"$X%"%O8"%4S"&G?""&&3O"'c#3""'pZK"'3V(*"(DH#H"(w(o()")HjIE")))*=*y"*l pvO#"*'*3F5("+*o738#"+o!QCR'F][gl<&Fdo,(FeqF^vFfuF+F;F+/F_pFcuF[gx,&F\vF+Fgcp#F gimF2,$FgcpF]]glFavQ,EIS~A052909F$"$*))*.Ff]lF;FdcmF;FeazF;,&F;F+Fe]pF+F;FgbelF ;F[_dlF;7#-F>6,Fhcm7%FB<#/FB-FL6(Fd^lF^dmF^bz-Fe^l6#-FLFj^lFccelFe_dlFis77F+F+F +F5F5F4FdpFhpF2FiqF]blFjqFcdmFabnFc\nF`crFhjpFj`mF[amF\`nFejrFa]gl&F+,$F[asFecglQjnnumber~of~ways~to~make~n~cen ts~with~coins~of~1~3~5~7~9~11~centsF$Q,EIS~A008675F$"$Y#*&F*F4,(F+F+F*FboF9F+F; 7#-F>6,Fav7%FB<&/F_s-FVFhfal/Fbs-FLFes/F_\n-FV6$FBF_s/FB-FL6$FbsF_\nFis77FinFin FinF+F5FdpFe]lFjqFf`mFbbnF\\lFfinF^_rF_jn"$w$"$4'"$')*Fdeal"%$e#"%!=%"%knFfdgl< 'F_uF[pF^pFi\l,*FeqF;FfuF;FjzF+F;F+-F^cl6$,$*&,(F7F+FfclF2FcclFe]lF+FhclF+Fh]n/ Fccl-F-6#,(F+F+F0FboFj]oF+*(Fa\_lF4,&Fa\_lF5*$Fa\_lF4F^vF;Fg\_lF+FavQ,EIS~A0000 71F$Fh[m-F_jl6#,$*$,&F;F+-F_jl6#,$*$,&F;F+-F_jl6#,$*$,&F;F+-F_jl6#,$*$,&F;F+Faf mF+F;F;F+F;F;F+F;F;F+F;F;7#-F>6,Q(MobilesF$7%FB<(/F[[p-FI6#F_[p/F\jo-FI6#F`jo/F djo-FI6#F[[p/F`jo-FI6#Fdjo/F_[pFN/FB-FI6#F\joFgn77FinF+FhpF[am"$S%"%qh"':`5"(51 7#")qCC\"+&>LhH"",&\;A:Q".!QuVFU7"/B>s*3XV%"1aQEC()4W%";Q@[a@uNE=?moD"==w#eI=I=UHhTuc"F ]gglF^rF^r*&F]qF;-Fc[l6#,$*&,(-F_jl6#,&-Fc[l6#*&,&-Fc[l6#*&,&-Fc[l6#*&FbfzF+Fh[ yF+F+F;F+F+-Fc[l6#,$F\\hlF;F+F+F;F+F+-Fc[l6#,$Fh[hlF;F+F+F;F+F+F;F+F`\hlF+F+F]q F+F;F+Q4Mobiles~of~height~5F$Q,EIS~A000359F$"$,$-F-6#,(F0F+-Fc[l6#*&-Fc[l6#F0F+ F*F+F;F+F+7#-F>6,F^hl7%FB<%/FB-FFFh_alFbhxF`jxFgn77FinF+F4F\_lFdfw"%wD"&$oW"'x> %*")D!)QB"*j,_o'",6J*Hi@"-S#3**y!y"/8u!4[N6$"1@ONXa'*e8"2zt*)=5@SW'"4rf:n\42))H $"6(\;w+Nu\A8="8!QC$)e7]"=8^1""9zc%379#H&\;)em";\d&z'RDA5&y4WT%"=T@2P(Rk0xsnWK4 $Ff\hlF^rF^rF^rFavQ,EIS~A052888F$"$j)*&FafmF+F\hglF+7#-F>6,F^hl7%FB<%Fb`yF\[`l/ F_sFbdwFgn77FinFinF5F:FeclFdbr"%(\#"&%RB"'KTD"(E)\J")a`%R%"*GSJ#o",OS/w;""-?:<- %=#".S=d&)QV%"/7hPh*)4(*"1_2H"*eJFK%"7[k>_* )**>nk"H"Fh^hlF^rF^r*(-F_jl6#,$*$,&F;F+*&FbfzF+Fg[yF;F+F;F;F+F]qF;-Fc[l6#,$*&,& -F_jl6#FbfzF+F;F+F+F]qF+F;F+QGProduct~of~Cycles~and~Cycles~of~CyclesF$Q,EIS~A05 2826F$Fedr,$*&Ff]lF+,*F+F+F*F^vFjxF;F9F+F;F;7$-F>6,Ffan7%FB<#/FB-Fe^l6#-FL6$FNF agdlFis77F+F5Fe]lFg`mFc_s"$G#"$L("%cB"%tv"&UV#"&V#y"')\^#"'&R3)"(S%)f#"(6,F`el7%FBFfahlFgn77F+F 5Fb\lFd\l"%/<">#"'gx_")SU(=""*gLM0$"+g\AL))"-+%)>GRG"/+kO&**Q+""0+?V)\AsQ"2+? bF*R0=;"3+/f"z0?8G("5+!G")oCit1^$"7++7P%=gS3b!="8+S5vdUwB=f')*":+!['R0c7+%))>3d "<+S#)HCPi3Q[:h[$">++KW"\Iw,Ylz4TAF_ahl<&Fdo/F_pFb\lF]am,*F`[_lF+*&,(F\qF;F]qF\ cuF1F+F+FfuF+F+*&,&F]qF^vFdqF+F+FjzF+F+F[rF+*&F`chlF+F_glF+FjchlFavQ,EIS~A05264 1F$"$(e,$*(,(F5F+F*F\cuFjxF5F+Ff]lF;Fc[nF;F;7#-F>6,Ffan7%FB<#/FB-FV6$Fd^l-Fe^l6 #-FV6$Fb[elFNFis77F5F4F2Fb\lFddmFb^mFj^mF\gblF^gblF`gbl"&Z4""&e'G"&E]("'>k>"'IU ^"(qiM""(zX_$"(muA*")=y:C")()fCj"*U,el"Fgfhl<&FdfnFd\xFibn,*FeqF;FfuF4FjzF;F;F+ ,&F+F+-F^cl6$,$Fbe^lFbjyFi]nF+,$**,(F5F+-F-6$F\^n$"+8,m>QFgdlF\cu*$F[ihlF5F5F+, &F;F+F[ihlF+F;,&F[ihlF4F_ihlFboF;),&F4F+F[ihlF;F]qF+F;FavQ,EIS~A032908F$"$>*,$* &Ff]lF+,*F+F+FjxFboF[alF+F*F;F;F;7#-F>6,Ffan7%FB<#/FB-Fe^l6#-FL6$-FV6$-FL6$Fg[g lFNFNFNFis77F+FinF5F5FhpF:FabnFbbnF\]sFahq"$E#"$J%"$@)"%k:"%!)H"%xc"&;3""&11#"& e#R"&$zu"'$\U"Ffihl<'FdoFdflF[pFac[l,*FeqF+FjzFboF[rF;FjblF+-F^cl6$,$*&,*FcclFh jpFdclFh`mFfcl!#wF`]lF+F+FhclF+Ficl/Fccl-F-6#,*F+F+FjemFboF_dlF+F0F;,$*(,&F;F+- F-6$F^\il$"+()f))[_FgdlF+F+,(*$Fb\ilF5FdpFb\ilF+*$Fb\ilFdpF[yF;),(F+F+Fb\ilF5*$ Fb\ilF4F;F]qF+F;FavQ,EIS~A052969F$"%T5,$FiawF;7#-F>6,F`el7%FB<#/FB-FL6$FN-FV6$F d^lFbipFgn77FinF5F5FjqFcelFhyFhgnFg`vFignF]^`lFjgnF_^`lF[hnFff_lF\hnFb^`lF]hnFd ^`lF^hnFf^`lF_hnF^]il<&FdflF_uF]am,&*&,(FboF+F\qF;F]qF^vF+FeqF+F+FjzF+*&-F^cl6$ ,$F[cwFfrF]cwF+F_glF+,&Fb^nF+Fb[lFfrFavQ,EIS~A052612F$"$d&,$*&Ff]lF5,*F;F+F*F4F 9FboF[alF+F;F;7#-F>6,Ffan7%FB<#/FB-Fe^l6#-FL6$-FV6$Fd[elFNFNFis77F+F+FdpF]blFhj pFijp"$C#F_du"%T<"%b["&TN""&lx$"'E`5"'^PH"'k#>)"(0\G#"(RDP'")SGx<")uzc\"*\PCQ"" *1"ebQFe^il<'FdoF^uFa[q,*FeqF+FfuFboF[rF4FjblF;/FioF]bl-F^cl6$,$*&,*F`_lF+FcclF gaoFfcl!$L"FdclF[amF+FhclF+#F+"$V'/Fccl-F-6#,*F;F+F0F4Fj]oFboF_dlF+,$*(,&F;F+-F -6$Faail$"+C!\be$FgdlF+F5,(FeailF^v*$FeailF4F2*$FeailFdpF[yF;),(F4F+*$FeailF5Fb oFjailF+F]qF+F;FavQ,EIS~A052946F$"%05-F^cl6$*(FddpF+FjdpF;-F_jl6#,$*$F[dwF;F;F+ Fbep7#-F>6,Q7Necklaces~of~2~coloursF$7%FB<%FgfpFhfp/FB-FI6#-FV6$F`fpFafpFis77Fi nF5F4FdpF2FiqFb\lFf`mF_blFihpFc^mFccoFffdl"$K'"%#=""%#>#"%;T"%7x"&-Y""&'fF"&)[_ FabilF^rF^rF^rQhoNecklaces~of~2~colours,~binary~irreducible~polynomials~whose~d egree~divides~nF$Q^oEIS~A000031.~http://sue.csc.uvic.ca/|ircos/inf/neck/Necklac eInfo.htmlF$F5*&F*FdpFafmF+7#-F>6,F^hl7%FB<$F^gm/FB-FL6'FNFNFNFNF_sFgn77FinFinF inFinFinFcenFa_mFif[lFg`v"&wD("'+[g"(+Cq&")+_()f"*+7*=p"+?"Hyr)"-+!)[z)="".+S#e cV<"/+?Rr0OF"0+?b2CJd%"1+))egLn4")"3++/6bPc?:F^dil<)F_uF[pF^pFapFfavFcfy,&*&,(F \qF;F]qF4FdpF+F+FeqF+F+Fji_lF+F^rFj^`lFavQ,EIS~A052778F$Fgd_l-Fc[l6#,&F*F+F9Fj_ w7#-F>6,Q+Cycles~setF$7%FB<#/FB-FF6#-FV6$-FI6$FN/FSF+-FI6$FNFfnFgn77F+F+F+F4F:F c\nFd]q"$^$"%L7"%pd"&T5$"'6?9"',mx"(8S([")Hx-F"*6"p$o""+T=">>""+,ocyw",$*\'\Z`" -R$))**>=%"."GM)pU/$Fieil<&FdoF^u,(F\^ilF+F[rF+FjzF;FgblF^r,$*0F[vF\vF5FfrF4Ffr F]qF\v-Fc[l6#*$F]qFj_wF+-Fc[l6#F[ablF+)F]q,$F]q#F;F4F+F_emQ]ppermutations~Sigma ~such~that~Sigma^3=Id,~degree~n~permutations~of~order~dividing~3F$Q,EIS~A001470 F$Fcel,$*(,(FboF+F*F4FjxF+F+Ff]lF;,(F;F+F*F5FjxF+F;F;7#-F>6,Ffan7%FB<#/FB-FV6$F d^l-Fe^l6#FcjblFis77F5F4F2FddoFi`mFc_sFb^wF]cyFjegl"%zB"%Ud"&hQ""&iM$"&$y!)"'E] >"'L3Z"(!pO6"(6Uu#"(5^i'")HW*f"")mRhQFihil<&FdfnFd\xFibn,*FeqF;FfuFboFjzF+F5F+, &F+F+-F^cl6$,$Fbe^lF`[l/Fccl-F-6#,(F;F+F0F5FjemF+F+,$**,(FboF+-F-6$F][jl$"+Cc8U TFgdlF4*$Fa[jlF5F+F+,&F;F+Fa[jlF+F;,&Fa[jlFboFe[jlFboF;),&F5F+Fa[jlF+F]qF+F;Fav Q,EIS~A052937F$"$G**&F]_oF+,(F;F+F^_oF+F*F+F;7#-F>6,F`al7%FB<#/FB-Fe^l6#-FL6$FN Fg_oFis77F+F+F+F+F+F+F+F+F+F5F4FdpFhpF2Fe]lFiqF:FgimFb\lFd]mFcdoF\\jl<,FdoF^uFg blFi\lFhblFcbzFicel,(FeqF+Fe`oF+FjbpF;FjcelF\del-F^cl6$,$*&,2Fccl!'\w6Fddu"([c2 "Feao"(7$H7Fdao"(G\S"Fa]o"(Kcg"Fdcl"(3]$=FfclFciel"'#>T*F+F+FhclF+#F+")f2g6,Fhcm7%FB<#/FB-FL6*Fd^lFbipFh alFhipF[jpF_jp-Fe^l6#-FL6\qFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFN FNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNF NFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFN-Fe^l6#-FL6drFNFNF NFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFN FNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNF NFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFi s77F+F+F5F5FdpFdpF2F2F:F:FjqFjqFabnFabnFg`mFg`mFhjpFhjpF_blF_blFedmFj^jl#Fdeq/-Ffo6#F^jnFieq/-Ffo6#FgfclFieq/-Ffo6#F`\vF^fq/-Ffo6#F\ `lF^fq/-Ffo6#Fe_ilFbfq/-Ffo6#Ff\`lFbfq/-Ffo6#FhjhlFgfq/-Ffo6#Fg^yFgfq/-Ffo6#F]b hlF[gq/-Ffo6#"$H#F[gq/-Ffo6#"$I#F_gq/-Ffo6#FghvF_gq/-Ffo6#FdcoFg]r/-Ffo6#Fi]clF g]r/-Ffo6#"$y#F[^r/-Ffo6#"$z#F[^r/-Ffo6#Fg\`lF`^r/-Ffo6#"$"GF`^r/-Ffo6#Fc^vFe^r /-Ffo6#FjclFe^r/-Ffo6#Fee[lFj^r/-Ffo6#"$&GFj^r/FfbrF\\l/FfirFacr/F[jrFacr/F_jrF ecr/FcjrFecr/FhjrFicr/F\[sFicr/F`[sF]dr/Fe[sF]dr/Fi[sFadr/FjbrF__l/F^crF__l/Fcc rFijp/FgcrFijp/F[drFjjp/F_drFjjp/FcdrF[[q/FgdrF[[q/F^`rFjq/Fd`rFabn/Fj`rFg`m/Fc arFhjp/F`^qFegr/Fd^qFegr/Fi^qFigr/F^_qFigr/Fb_qF^hr/Ff_qF^hr/F[`qFbhr/F^\sFadr/ F[erF\\o/F_erF\\o/FcerF\[q/FgerF\[q/F[frF][q/F_frF][q/FcfrF^[q/FgfrF^[q/F[grF_[ q/F_grF_[q/FcgrFdco/FggrFdco/F[hrFdbr/F`hrFdbr/FdhrFhbr/FhhrFhbr/F]irF\cr/FairF \cr/F_brFedm/FbbrF\\l/Fb\sFedr/Ff\sFedr/Fj\sFidr/F_]sFidr/Fc]sF]er/Fg]sF]er/F[^ sFaer/F`^sFaer/Fd^sFeer/Fi^sFeer/F]_sFier/Fa_sFier/Ff_sF]fr/Fj_sF]fr/Fc[qFafr/F h[qFafr/F]\qFefr/Fa\qFefr/Fe\qFifr/Fj\qFifr/F^]qF]gr/Fb]qF]gr/Fg]qFagr/F[^qFagr /F]\rF^^s/Fb\rFb^s/F[]rFb^s/Fg\rFg^s/F_]rFg^s/Fd]rF[_s/Fi]rF[_s/F]^rF__s/Fb^rF_ _s/Fg^rFd_s/F\_rFd_s/Fa_rFh_s/Fe_rFh_s/F`^]lF\`s/F]^]lF\`s/Fj]]lFf[q/Ff]]lFf[q/ Fc]]lF[\q/F`]]lF[\q/F\]]lF_\q/Fdh\lF_\q/Fah\lFc\q/F]h\lFc\q/FfarF_bl/FiarF_bl/F \brFedm/FidqFfjr/F^eqFjjr/FbeqFjjr/FfeqF^[s/F[fqF^[s/F`fqFc[s/FdfqFc[s/FifqFg[s /F]gqFg[s/FagqF\\s/FfgqF\\s/F[hqF`\s/F_hqF`\s/FdhqFd\s/FhhqFd\s/F]iqFh\s/FbiqFh \s/FgiqF]]s/F[jqF]]s/F`jqFa]s/FdjqFa]s/FhjqFe]s/F\[rFe]s/F`[rFi]s/Fe[rFi]s/Fi[r F^^s/-Ffo6#FccoF`aq/-Ffo6#FdfwF`aq/-Ffo6#FfhvFdaq/-Ffo6#FaboFdaq/-Ffo6#F]wFiaq/ -Ffo6#"$$>Fiaq/-Ffo6#F_ealF]bq/-Ffo6#F`[`lF]bq/-Ffo6#F\jnFbbq/-Ffo6#Fd`elFbbq/- Ffo6#Fh^mFgbq/-Ffo6#"$+#F[cq/-Ffo6#Fi\vF[cq/-Ffo6#Fa[`lFi\r/-Ffo6#F[^vFi\r/-Ffo 6#FhimFb]r/F][]lFd_q/Fij\lFi_q/Ffj\lFi_q/-Ffo6#Fg^mF]`q/-Ffo6#FdblF]`q/-Ffo6#"$ #=Fa`q/-Ffo6#F][qFa`q/-Ffo6#F[jnFf`q/-Ffo6#F]d^lFf`q/-Ffo6#FhdwF[aq/-Ffo6#Fi_lF [aq/-Ffo6#"$f#F^[r/-Ffo6#"$g#Fc[r/-Ffo6#Fd\vFc[r/-Ffo6#FghelFg[r/-Ffo6#F[fxFg[r /-Ffo6#"$k#F[\r/-Ffo6#Fc[zF[\r/-Ffo6#Fee_lF`\r/-Ffo6#"$n#F`\r/-Ffo6#"$o#Fe\r/-F fo6#"$p#Fe\r/-Ffo6#F\_mF]]r/-Ffo6#F`ealF]]r/-Ffo6#FbhelFbjq/-Ffo6#Fh_\lF`iq/-Ff o6#FedglFeiq/-Ffo6#F^`lFeiq/-Ffo6#"$[#Fiiq/-Ffo6#Fe_vFiiq/-Ffo6#FeblF^jq/-Ffo6# F\`vF^jq/-Ffo6#F[_mFbjq/-Ffo6#"$a#Ffjq/-Ffo6#FjgalFfjq/-Ffo6#F_e_lFjjq/-Ffo6#Fe bnFjjq/-Ffo6#"$e#F^[r/-Ffo6#"$8#Fgdq/-Ffo6#"$(H"'3kX/-Ffo6#"$)H"'C;Z/-Ffo6#"$v# Fb]r/-Ffo6#Fd[[lF]dq/-Ffo6#F^[qFbdq/-Ffo6#F[`lFbdq/-Ffo6#F_gelFgdq/-Ffo6#"$9#F\ eq/-Ffo6#"$:#F\eq/-Ffo6#Fi^mF`eq/-Ffo6#FjczF`eq/-Ffo6#"$=#FdeqFj_dl/F\bpF:/-Ffo 6#Fd_vF__r/-Ffo6#F]_mFc_r/-Ffo6#FfddlFc_r/-Ffo6#"$!HFg_r/-Ffo6#"$"HFg_r/-Ffo6#F b]y"'#yE%/-Ffo6#"$$HFab\m/-Ffo6#Fi]_l"'#>T%/-Ffo6#FcdzFib\m/-Ffo6#F^]pFd^\m/-Ff o6#"$'GF__r/F]arFg`m,dam!3+g(p]Z-WM#F+F]q!1?"R&>tM;aF``s!%@@FeqFeenFfu"&!GNFjz" '!3O"Fcq!*NKht*Fe\o"(S)p=F[r"'!3)QFjbl"'SA"*Fb`o"(![iMF\q!/kLO["HO&F`q!-COQ)*\H F[as!(vz#>F]as"++W(>V%Faas"+?F]XZFeas"+S#e+8%Fias"++C1hNF]bs"+SUvRQFabs"+S)fYz# Febs"++_^QIFibs"+?F)RH$F]cs",+C%\"Q"Facs",![!e&H8Fecs",![%[vA"Fics",glE#y7F]ds" ,!3=dx6Fads",!on"*z5Feds",+7X$G6Fids"++3Xc)*F]es",?jMB."Faes"+?f:^*)Fees"++'>*) R*Fies"+?BM'3)F]fs"+SE_8&)Fafs"+?*R,(oFefs"++#HWE(Fifs"++s*)pwF]gs"+S1>(['Fags" +?Bq:hFegs"++cvbdFigs"++Kkq]F]hs"+SeR2aFahs"*+S1(>Fehs"*g`t"HFihs"*gP8T#F]is")g t?**Fais"*SA/f"Feis"*SYeE"Fiis"+g0!>T"F]js"+?j/>6Fajs"+!)[)*f7Fejs"*gDZf(Fijs"* gD4p)F][t"*!3?'f'Fa[t"*+'pv[Fe[t"*?>8p&Fi[t"*S%pWTF]\t"*SEN\$Fa\t"++cyTBFe\t"+! ohF8#Fi\t"+?ZKN>F]]t"+g\*\d"Fa]t"+g$H%\ F%Fb^t")+%ex&Ff^t")!oH3$Fj^t")Smi@F]cp")S9n9Fjbp"(!)3c*Fe`o"(!3KfFa_t")glUwFe_t ",+w`&[;Fi_t",?>.Jq"F]`t",!G$>Vf"Fa`t",gxZ/a"Fe`t",!)[()p["Fi`t",![o)RV"F]at",g (f'f9#F`at",gTPY.#Fcat",S-!G!4#Ffat",S!pcB>Fiat",?Je!z>F\bt",S9z;?#F_bt",?V2uD# Fbbt",gxUJJ#Febt",Su&4ODFhbt",+sM_O%F[ct",!3!>YU#F^ct",gPd.[#Fact",+k!))oBFdct" ,+wx4T%Fgct",+OZNe%Fjct",?6i?a%F]dt",+%)R%*\%F`dt",!G*GdX%Fcdt",+)[LBTFfdt",!3# QDA%Fidt",+_6L<%F\et",?ncE2%F_et",+GD8-%Fbet",?">p#[$Feet",SU;tU$Fhet",?>[&=VF[ ft",+cm4F%F^ft",SA>z`$Faft",++klv$Fdft",!o&HBq$Fgft",g8!yZOFjft",?2iHf$F]gt",+[ '*o"RF`gt",S=/#\JFcgt",!)3*QpRFfgt",+%e*Q'QFigt",?bN/"QF\ht",g,D=P$F_ht",?"fC;L Fbht",SI.1E$Feht",?N<\?$Fhht",+[svk#F[it",?&f_qGF^it",Se(y9GFait",g@\!fFFdit",! [3J.FFgit",?bSx.$Fjit",?6M=f#F]jt",g*eZ$4$F`jt",!)o-?)HFcjt",+Kki#HFfjt",S5kH"= Fijt",gT">o=F\[u",gD@zv"F`[u",+![AsZFc[u",?V]rt%Ff[u",+7(p+ZFi[u",![K"Hm%F\\u", +?ZQi%F_\u",+ort*\Fb\u",!)oq](\Fe\u",+o.5&\Fh\u",+/>_#\F[]u",?:lx*[F^]u",+S!po[ Fa]u",!oJ/Q[Fd]u",+%=(e![Fg]u",g8$***4&F[^u",![))Q"4&F^^u",!oRp!3&Fa^u",!3y%z1& Fd^u",+s%>`]Fg^u",+3#[O]Fj^u",?Fey,&-Ffo6#,&F]qF+F_[qF+Fde\m-Ffo6#,&F]qF+F`ijlF +Fbh\m-Ffo6#,&F]qF+F\ijlF+Fah\m-Ffo6#,&F]qF+F]bhlF+F_h\m-Ffo6#,&F]qF+Fg^yF+F`h\ m-Ffo6#,&F]qF+F^\\mF+Fhe\m-Ffo6#,&F]qF+F^`lF+Fie\m-Ffo6#,&F]qF+FedglF+Fje\m-Ffo 6#,&F]qF+Fh_\lF+Ffe\m-Ffo6#,&F]qF+F]c`lF+Fbe\m-Ffo6#,&F]qF+FdcoF+F\g\m-Ffo6#,&F ]qF+F]_\mF+F]g\m-Ffo6#,&F]qF+FhimF+F[g\m-Ffo6#,&F]qF+F[^vF+Fif\m-Ffo6#,&F]qF+Fa [`lF+Fjf\m-Ffo6#,&F]qF+F`ealF+Feg\m-Ffo6#,&F]qF+F\_mF+Fgf\m-Ffo6#,&F]qF+Fhj[mF+ Fhf\m-Ffo6#,&F]qF+Fdj[mF+Fff\m-Ffo6#,&F]qF+F`j[mF+Fcg\m-Ffo6#,&F]qF+Fee_lF+Fdg\ m-Ffo6#,&F]qF+Fc[zF+Fbg\m-Ffo6#,&F]qF+Ffi[mF+Fag\m-Ffo6#,&F]qF+FghelF+Ffg\m-Ffo 6#,&F]qF+Fd\vF+Fhd\m-Ffo6#,&F]qF+Fih[mF+Fid\m-Ffo6#,&F]qF+Feh[mF+Fjd\m-Ffo6#,&F ]qF+F`jnF+Fge\m-Ffo6#,&F]qF+FhyF+Fce\m-Ffo6#,&F]qF+Fc^\mF+Fjc\m-Ffo6#,&F]qF+Ffd dlF+Fjg\m-Ffo6#,&F]qF+FhdwF+Fgj\m-Ffo6#,&F]qF+F[jnF+Fji\m-Ffo6#,&F]qF+F][qF+Fgi \m-Ffo6#,&F]qF+Fbg[mF+Fii\m-Ffo6#,&F]qF+FdblF+Fhi\m-Ffo6#,&F]qF+F\jnF+F]j\m-Ffo 6#,&F]qF+F`[`lF+F\j\m-Ffo6#,&F]qF+F_ealF+F`j\m-Ffo6#,&F]qF+Ffd[mF+Fdj\m-Ffo6#,& F]qF+F]wF+Fcj\m-Ffo6#,&F]qF+FfhvF+Faj\m-Ffo6#,&F]qF+FdfwF+Fij\m-Ffo6#,&F]qF+Fcc oF+Fhj\m-Ffo6#,&F]qF+Fi_lF+Fej\m-Ffo6#,&F]qF+F]jnF+Fa[]m-Ffo6#,&F]qF+F^fmF+F`[] m-Ffo6#,&F]qF+F[cjlF+F_[]m-Ffo6#,&F]qF+F\cyF+Fg[]m-Ffo6#,&F]qF+FchdlF+Ff[]m-Ffo 6#,&F]qF+FabjlF+Fc[]m-Ffo6#,&F]qF+Fi\vF+Ffj\m-Ffo6#,&F]qF+Fie[mF+F][]m-Ffo6#,&F ]qF+F]_mF+Fig\m-Ffo6#,&F]qF+F[^\mF+Ffd\m-Ffo6#,&F]qF+FebnF+Fgd\m-Ffo6#,&F]qF+Fj galF+Fcd\m-Ffo6#,&F]qF+F^]\mF+Fdd\m-Ffo6#,&F]qF+FbhelF+F^f\m-Ffo6#,&F]qF+F[_mF+ F_f\m-Ffo6#,&F]qF+F\`vF+F]f\m-Ffo6#,&F]qF+FeblF+F\f\m-Ffo6#,&F]qF+Fe_vF+F[f\m-F fo6#,&F]qF+FajjlF+Faf\m-Ffo6#,&F]qF+Fi]clF+F^g\m-Ffo6#,&F]qF+Fia\mF+F[h\m-Ffo6# ,&F]qF+Fd_vF+Fgg\m-Ffo6#,&F]qF+Fcc\mF+Fhg\m-Ffo6#,&F]qF+Fe[[mF+F^h\m-Ffo6#,&F]q F+Fee[lF+Fcf\m-Ffo6#,&F]qF+FjclF+Fef\m-Ffo6#,&F]qF+Fc^vF+Fdf\m-Ffo6#,&F]qF+Fhjj lF+F`f\m-Ffo6#,&F]qF+"$*HF+Feen-Ffo6#,&F]qF+Fh^\mF+Fic\m-Ffo6#,&F]qF+F^]pF+F]d\ m-Ffo6#,&F]qF+FcdzF+F^d\m-Ffo6#,&F]qF+Fi]_lF+F\d\m-Ffo6#,&F]qF+Feb\mF+F_d\m-Ffo 6#,&F]qF+Fb]yF+F]h\m-Ffo6#,&F]qF+F[fxF+F`g\m-Ffo6#,&F]qF+FcfjlF+Fee\m-Ffo6#,&F] qF+FiejlF+F`e\m-Ffo6#,&F]qF+F_^\mF+F`i\m-Ffo6#,&F]qF+Fj_lF+F\[]m-Ffo6#,&F]qF+Fh ^mF+F[[]m-Ffo6#,&F]qF+F]`\mF+F_i\m-Ffo6#,&F]qF+F_gelF+F]i\m-Ffo6#,&F]qF+F[`lF+F d[]m-Ffo6#,&F]qF+F^[qF+F^[]m-Ffo6#,&F]qF+Fd[[lF+Fb[]m-Ffo6#,&F]qF+FhjhlF+Fj[]m- Ffo6#,&F]qF+Ff\`lF+Fh[]m-Ffo6#,&F]qF+Fe_ilF+Fi[]m-Ffo6#,&F]qF+F\`lF+Fhh\m-Ffo6# ,&F]qF+FgfclF+Ffh\m-Ffo6#,&F]qF+F^jnF+Fgh\m-Ffo6#,&F]qF+F]gjlF+Feh\m-Ffo6#,&F]q F+F[a\mF+Fjh\m-Ffo6#,&F]qF+FjczF+F[i\m-Ffo6#,&F]qF+Fi^mF+F\i\m-Ffo6#,&F]qF+Fa`\ mF+Fai\m-Ffo6#,&F]qF+Fhe[lF+Fae\m-Ffo6#,&F]qF+Fb\vF+F_e\m-Ffo6#,&F]qF+F_ejlF+F] e\m-Ffo6#,&F]qF+F]`lF+F^e\m-Ffo6#,&F]qF+Fj^mF+F\e\m-Ffo6#,&F]qF+F^]blF+F[e\m-Ff o6#,&F]qF+F_jnF+Fdh\m-Ffo6#,&F]qF+FghvF+Fch\m-Ffo6#,&F]qF+F^f[lF+Fe[]m-Ffo6#,&F ]qF+FaboF+Fbj\m-Ffo6#,&F]qF+Fd`elF+Fjj\m-Ffo6#,&F]qF+F]d^lF+F[j\m-Ffo6#,&F]qF+F `\vF+Fih\m-Ffo6#,&F]qF+F]jjlF+F_g\m-Ffo6#,&F]qF+F_e_lF+Fed\mF`g]lF[\]mFcg]lF\\] mFfg]lF[]]mFig]lFj\]mF\h]lFg\]mF_h]lFf\]mFbh]lFe\]mFeh]lFd\]mFhh]lFc\]mF[i]lFb\ ]mF^i]lFa\]mFai]lFdi\mFdi]lFci\mFgi]lF_\]mFji]lF^\]mF]j]lF]\]mF`j]lF_j\mFcj]lF^ j\mFfj]lF^i\mFij]lFbi\mF\[^lFfi\mF_[^lFei\mFb[^l",gduk5&Fe[^lFi\]mFh[^lFh\]mF[\ ^lF_c^mF^\^l",g(*336&Fa\^l",!3-)H6&Fd\^lFac^mFg\^lF`c^mFj\^lF`\]mF]]^lF^]]mF`]^ lF]]]mFc]^lF\]]m-Ffo6#,&F]qF+Fg\`lF+Fbf\m-Ffo6#,&F]qF+F]b\mF+F\h\m,`oF]q#".()[> b#)f$".?B*>F0xF``s#"$2$".+'p&)eM=Fcq#"(zt=&"-?Rr5Ff[\lF+-F^cl6$,$*.,2FcclF+FdclFdpFfcl F+FdduF5Fa]oF5FdaoF5FeaoF5FboF+F+FjbuF+FhuF+F\[lF+F]rF+F\clF+#F;",!oN&)RMF[h^mF +-F^cl6$,$*.,&FcclFf_l!$]"F+F+FjbuF+FhuF+F\[lF+F]rF+F\clF+#F+",?B&H:hFf[\lF+-F^ cl6$,$*&,RFccl!#$*FdclFh[mFfclFhbvFddu"%a7Fjdu"%[7Fa]oFh[mFdao"$D)FeaoFbixFbfuF h[`lF\eu"%Q;Fdfu"%U@Fgfu"%y=FefuF\][lFafu"%;?Fbdu"%Z;Fifu"%26Fcfu"%u?F`^v"%K;Fh du"%G?Fffu"%"*=F[guF\gnFjfu"%)4"FhfuFe`y!$w&F+F+FhclF+F]f^mF]guF+-F^cl6$,$**,&F ccl"()3'*\!(8<([F+F+F[huF+FhuF+F\[lF+#F;Ffi^mFf[\lF+-F^cl6$,$*.FcclF5,6FdaoFhpF eaoFdpFfclFdpFdclFdpFa]oFhpFjduF4FdduFdpFbduF4FcclF4F4F+F+FccuF+FhuF+F\[lF+F]rF +#F;"*?*4qjFaeuF+-F^cl6$,$*(,&Fccl"+h'G(*=""+T&*oR7F+F+F]buF+FhuF+#F;"-+;wkdIFf [\lF+-F^cl6$,$*&,:Fccl!)9j_6Fdcl!)`yP7Ffcl!))[_7"Fddu!(NyU%Fjdu!(/RG%Fa]o!)_^17 Fdao!(i+C)Feao!(;Q,)F\eu"(oPO$Fbdu"(yJT$Fhdu")"4'\5!)?VE5F+F+FhclF+#F+Fj[_mFaeu F+-F^cl6$,$**,2Fdcl!'HJ5Fddu"'0p9Fa]o!')RY#Fdao!''QG#Feao!'%*[AFfcl!'wK\Fccl!'[ 0]!'V\9F+F+F[huF+FhuF+F\[lF+#F+F^i^mF[h^mF+-F^cl6$,$*0F^j[lF+)Fccl,&F1F+F]qF;F+ FhuF+F\[lF+F]rF+F\clF+Fg\oF+Fb\_mFf[\lF+-F^cl6$,$*&,2Fccl!*"=XUiFdcl"+U[F6JFfcl !*wX>*eFddu"+%)Gp(*\Fdao"+,**>\=Feao"+c-*Qy"Fa]o!+#*4#3!=!+%QSiz%F+F+FhclF+#F+" -+%yE*>"oF"Feao")H>18Fdao!)QEm;Fa]o!)5%>r"Fdcl!(jQ! oFfcl!(@,,(Fccl!)/%*GR!)+0HSF+F+F]buF+FhuF+Fc^_mF[h^mF+#!/t))GegH_F^e^mF+-F^cl6 $,$*(,:Fdcl!'A+FFfcl!'EQFFddu!'aFAFjdu!'JR9Fa]o!',FJFdao!'T"3$Feao!'YiAF\eu"&5% \Fbdu!'$[T"Fhdu"&'3\Fccl!'Z(G#!'r6,Q1Non-planar~treesF$7%FB<$/F_s-FVFeicl/FB-FF6$F_s1 F4FSFgn77FinFinFinF+F+FgimF_blFaeal"%6B"&_i#"'$*QC"("GtJ")zo"*Q"*=C#Re"+iU")3)) "-ZgZI::".cVReYp#"/N+,[wg_"1f$ojtvH2""28[Z'G4g^B"3O];k)3#Q)R&F\g_mF^rF^rF^rQhnN on-planar~labelled~trees~with~neither~unary~nor~binary~nodesF$Q,EIS~A052526F$Fh aq,$*&F*F5Ff]lFboF57#-F>6,F`el7%FB<#/FB-FL6&FNFd^lFd^lFc`mFgn77FinFinFdpFcelFe^ m"$g*"%+s"&![gFfhbl"(!31e")+%=`'"*+gL)z",+_.Q0""-+#*\[%\"".+7dNmE#"/+SI#)[hO"0+ +k'p$oF'"2+?2*p(*>Q6"3+?v%*fq!o<#"4+?&zr9OAzV"5++K72Jw-X#*F\i_m<&F_uF[pFa[q,&*& ,&F\qF;F]qF;F+FeqF+F+*&F^hoF+FfuF+F+*&,&F]qF5FboF+F+F_glF+F_amFavQ,EIS~A052609F $"$a&,&*$FfglF5F+FfglF;7#-F>6,Fav7%FB<#/FB-FL6$FhhlFg\alFgn77FinF+F4Fe]lFcdmF]_ lFhdmFd_lFjgalFecrF[halF\halF]halF^halF_halF`halFahalFbhalFchalFdhalFehalF_[`m< %F^uF_u,(FjzF+FeqF5FfuF^vF^r,$*,F[vF\vF5FfrF]qF\vFa^alF+FigyF+FfrFavFjhalFf]r-F c[l6#-F^cl6$,$*,Ff\wF+FaepF+,&FaepF+FboF+F+Fg\wF;FjdpF;F;Fbep7#-F>6,F^hl7%FB<%F cdwFa]el/FBF_]wFis77F+F5F5F4FdpFhpFe]lF:FgimFb\lFd]mFg`mF\`oFbbnF[amFcirFacoFid mF[emF_`nFa`nF^\`mF^rF^rF^rFavQ,EIS~A052816F$"$"y*&Fh\zF+FhiblF;7#-F>6,Ffan7%FB <#/FB-Fe^l6#-FL6$-Fe^l6#-FV6$FdipFNFc`mFis77F+F5F2Ff`mF]^sF[a\mFhgn"%yB"%ay"&Sf #"&uc)"'iHG"'gX$*"(Um3$")'[%>5")+,nL"*'y/76"*eWGn$"+g"eI@""+Q*ek+%",u\VKK"F^]`m <&FdoFdfnFf[clF]am-F^cl6$,$Ff]n#FboFddoF\\cl*(,(Fb\clF+Ffh^lFfrFd\clF+F+Fc\clF; Ff\clF+FavQ,EIS~A052991F$"%m5,$*$,(F;F+FjxF+F9F+F;F;7$-F>6,Fav7%FB<#/FB-Fe^l6#- FL6$FNF[j^lFis77F+FinF+F+F+F5F5F4FdpFhpFe]lF:FjqFcuFc\nFh`mF[`nFcirFcbnF[_qF[[q Fe_`m<&FdoF[pFgbl,(FeqF+FfuF+F[rF;-F^cl6$,$*&,(F^vF+FcclFa`lFfclF5F+FhclF+#F;Fc do/Fccl-F-6#Ffffl,$*&,&F]gflFbo*$FdfflF4F^vF;F[gflF+F;FavQ,EIS~A000931F$"$$R-F> 6,F`el7%FBF\``mFgn77F+FinF5F2FcelFhyFaiwF]g[l"'!Gh""(+W"=F[i_l"*+7Df$"++#>![d", +GLK'**".+_6W2$=F`j_m"0+g&eAVTx"2+SqwRoGu""3++Ks3HahT"5+?bf^jy9Y5"6++'p8K*G3Nx# Fe_`m<&FdoFdflF[p,(Fb[wF+F`fhlF+F[rF+*&Fe``mF+F_glF+F^a`mFavQ,EIS~A052597F$Fb]o Fidy7#-F>6,Fav7%FB<$Ff\al/FB-FL6%FNFNF_sFgn77FinFinF5F2FjqFf`mFi`mFc\lFb[sFa^mF b^mFheqFd\lF\[qFbg[mF^[qFhyFa[`lF^_mF`_m"$!QFidy<&FdflF_uF[p,&FbhnF+Fjj_mF+F^r, $*0F[vF\vF5FfrF]^alF+FcgyF+F]qFb^n-Fc[lFfqF+FigyF+FfrFavQ,EIS~A002378F$"$5%,&Fa fmF+FfglF+7#-F>6,F^hl7%FB<%F[gmF^gm/FbsFg\alFgn77F+F5F5F4Fe]lF`crFjhqF_\y"%T]"& @.%"'")GO"(,)GO"),o"*R"*,;+z%"+,3-Fi",,7Hyr)".,!oVn28"/,!)))*yA4#"0,g4Guob$"1,! GdqtBS'"3,?$)3/5X;7Fhc`m<'FdoFdflFhoF]am,(FijmF+*&,(F\qF;F]qF^vF;F+F+FjzF+F+*&F [rF+F]qF+F+F^rFj^`lFavQ,EIS~A038507F$"$M),$*&FieoF+Fc[nF;F;7#-F>6,F`el7%FB<#/FB -Fe^l6#-FL6$FN-Fe^l6#-FLFagzFgn77F+F+FdpFi`m"$7$F]`gl"&!3k"(?V<"")+_fC"*g$>&z&" ,+G,s^""-+GHHpV"/+7&)[ns8"0++_NA=n%"2+;#4g&QBr""3+?F#e#3WCn"5+?(ys>Mqn"G"7+!)[V P%R3[OD""8++_>a_'Gzw2f":+SuR')=BY2!oQH"<++/n*e;^[jGrQ:Fde`m<%FdoF^u,(*&,(F\qF+F 5F+F]qF4F+FeqF+F+Fa^xF+FjzF+*&-F^cl6$,$*&,&F;F+FcclFdpF+FhclF+FbjyFi]nF+F_glF+, $*(,&F5F+F_^nF;F+Fc^nF;Fe^nF+F;FavQ,EIS~A052574F$"$<&-Fc[l6#-F^cl6$**FaepF+Fd\` mF+Fg\wF;FjdpF;Fbep7#-F>6,Fav7%FB<%/FB-FFFchlF]eglF^]zFis77F+F5FdpFe]lFjqF\_lFi `mFedmF__lFjjpFe_lF`[`lFa[`lFb[`lFc[`lFd[`lFe[`lFf[`lFg[`lFh[`l"%9FFch`mF^rF^rF ^rFavFi[`lFffq*&F*F+-Fc[l6#-F^cl6$*(FaepF+-Fc[l6#-F^cl6$*&)FaepFd_xF+Fd_xF;F[`x F+FjdpF;FbepF+7#-F>6,F``x7%FB<%Fc`xFaax/Fi`xFNFis77FinF+F+F5F4FhpFe]lFgimFcdmFg `mFi`mFc\lFb[sFjdmFjhmF\\oFc[[lFghvFc^\mFd`p"$!\Fai`mF^rF^rF^rQ=Unlabelled~tree s~of~height~2F$Q,EIS~A000041F$F[\o*(F*F+F_hpF;Ff]lF;7#-F>6,F`el7%FB<#/FB-FL6%FN Fd^lFbipFgn77FinF+F5FjqFeclFa_mFdelFeelFha`mFia`m")+S9="*+3]R#"++'4S(G",+c9*eV" -+%Q![-hFhdfl"0+S5>BQn""1+S'G&o=,K"2+?b^LO@w&"4++K)3/5X;7"5++Sm<3?!HV#Fgj`m<%,( *&,*F`qF;F\qF\cuF]qF7F[yF+F+FeqF+F+FdawF+FjayF+F^uF_u*&,(F]qFfrF`[lF+)F;,&F+F+F ]qF;F`[lF+F_glF+Fi[lFavQ,EIS~A052591F$"$O&,$*&F*F+F[[mF;F;7#-F>6,F`el7%FB<#/FBF fhyFgn77FinF+FiqFhaq"%O:"&?2$"'!GP(")SQk?"*!)Ggg'",!o.$*y3O/&e"6+!3=ft_;E5N"8+?rv)4vducYA":+gT[:nq7(emF:"<+? &f[^$)[JF%>**4""=+?NKHr9JfZwQf$)"?++;)eMq<\u!=,^(o'Fe\am<%F^uF_uFi\m*&)FdpF^hoF +F_glF+,$F^]mF`[lFavQ,EIS~A034177F$Ffbn*&F\iilF;Ff]lF;7#-F>6,Ffan7%FB<#/FB-FL6$ FeiilFd^lFis77F+F4FiqFf`mFcirFahqF]_m"$'p"%"o""%fS"%+)*"&gO#"&@r&"'.z8"'GHL"'gP !)"(\/%>"(fYo%")o(48"")'>/t#")h"=f'Fd^am<%FdoFd\x,*FeqF;FfuFboFjzF+F;F+,&F\vF+- F^cl6$,$F[cwF`[lFjjilF+*(Fg[jlF;Ff[jlF;Fh[jlF+FavQ,EIS~A048739F$"%l5-Fc[l6#-F^c l6$**Ff\wF+FaepF5Fg\wF;FjdpF;Fbep7#-F>6,F^hl7%FB<%FcdwFbhxFj\`mFis77F+FinF+F+F+ F5F5F4F4FhpFhpFe]lFiqF]blFjqFcdmFabnFc\nF`crFhjpFddmFe`amF^rF^rF^rFavQ,EIS~A025 147F$"$)z,$*(F*F+,&Fa`lF+F*F4F+Ff]lFboF;7#-F>6,Fj]l7%FB<#/FB-FV6$F\balFe^zFis77 FinFe]lFgimFcdmF\_lFcdoF\`oF]_lFddmF]`oF^_lFjhrFfdmF`]lFacoFhdmF__lFc_sFj[qF`_l F]^qFbaam<&F_u,(FeqF+F^vF+F]qF[yF`jdlFadal,&F4F+F]qFdpFbbalQ)~4~n~+~3F$Q,EIS~A0 16837F$"$S*,$*(F*F+,&Fd[wF+F*F5F+Ff]lFboF;7#-F>6,Fj]l7%FB<#/FB-FV6$Fc_n-FL6$Fc` mFd^lFis77FinFgimFf`mFhjpF`rFjhrFb[sFcbnFe[qF]^qFe`qFjhmFheqFahqFgaoFa]rFj]uF_h \lFadvF\\]lFbg[mFdbam<&,(FeqF+FboF+F]qFdqF_uFf[[lFcdv,&F5F+F]qF:F[anQ)~9~n~+~2F $Q,EIS~A017185F$"$k*,$FdfmFbo7#-F>6,F`el7%FB<#/FB-FVFe[elFgn77F5F5FdpFjqFeclFhy FaiwFg`vFdg[lF]^`lF^^`lF_^`lF`^`lFff_lFa^`lFb^`lFc^`lFd^`lFe^`lFf^`lFcgdlFhcam< $FahnFibnFfgdlF5FavQ,EIS~A052553F$Fdj`m,$*&FafmF+F`hglF;F;7#-F>6,Fav7%FB<%Fb`yF \[`lF`hxFgn77FinF+F4Fb\lFfin"$%p"%yl"&#zs"'/4#*"))34J""*74O2#"+cIB3O",Sm[&\o".k !y>j39"/!3g;g(>J"0%=DH%QIS("2c-gT;BQ(="3'*3gp]fPR]"57p3'R0$**)\V""6;s8;DtMEKJ%" 8C%f^-Mm(=%ok8FcdamF^rF^r*,F``hlF+,&F+F+Fe`hlF;F;,&Fg[yF;F+F+F;Fd`hlF+Ff`hlF+Fa vQ,EIS~A007840F$F_iq-Fc[l6#,*FfglF+F;F+F*F;FjxF\v7#-F>6,Fav7%FB<#/FB-FF6#-FF6$F NF[h_mFgn77F+FinFinF+F+F+FgimF_blFe`q"$"\"%dD"&`8""&0,'"'1DO"(Y#p@"):3e8")Nu#>* "*(4y+l"+ZO-iZ",IN#*3l$"-!4(3<@HF_famF^rF^rF^rFavQ,EIS~A006505F$Fccq*&FdcmF;Fie oF;7#-F>6,Ffan7%FB<#/FB-FL6$F^dmFdfoFis77F+F5FdpF:Fd]mF_blF]`nFj]uFb]y"$&e"%q6" %SB"%"o%"%i$*"&C(="&\u$"&)*[("''z\""'$f*H"''=*f"(s$)>"Fjgam<&FdoFa[qF]am,,FeqFb oFfuF;FjzF;F[rF+F;F+,&F`ho#FiqFe]l-F^cl6$,$*&,(FdpF+FcclF+FfclF5F+FhclF+#F;Fc\n /Fccl-F-6#,&F;F+Fj]oF+F+,$F`hoFciamFavQ,EIS~A033138F$"$E**&,(F;F+F[alF;F\[oF+F+ Ff]lF;7#-F>6,F`el7%FB<#/FB-FV6$Fd^lFjalFgn77F+F+F5F2FeclFcenFhgnFeenFignFgenFjg nFienF[hnF[fnF\hnF]fnF]hnF_fnF^hnFafnF_hnFajam<)FdoF^uFdflFahnFcfyFdzFegdlF_glF +FavQ,EIS~A052596F$"$T&,$*$,(F;F+F*FhpFjxF+F;F;7#-F>6,Ffan7%FB<#/FB-Fe^l6#-FV6( FNFNFNFNFNFdipFis77F+FhpF^blF\\o"$,("%SO"&,*="&X")*"'E'4&"(vik#"),5u8")!G^8("*, u\q$"+&GQQ#>"+E))o*)**",:CGs=&"-,45^$p#".?pyF')R"".,b.!\is"/DW'z26x$"0Ew&)zy"e> F_[bm<%Fdo,(FeqF+FfuFhpFjzF;F[gx-F^cl6$,$*&,&FhpF+FcclF5F+FhclF+#F+Fhjp/Fccl-F- 6#,(F;F+F0FhpFjemF+,$*&,(#F`crF5F+*$FhjpFfrFb\cl*$,&Fb\clF+F^^bmFfrF5FboF;)F`^b mFfjlF+F;FavQ,EIS~A052918F$"$,*-F-6#,(F0F;*&F0FiqF*F+F+F+F+7#-F>6,F``x7%FB<$Fah l/F_s-FL6*FNFBFBFBFBFBFBFBFis77F+F+FiqFe`q"%S7"&y#="'%Q&G"([$QY")C?lx"+02*)H8", ?(H+>B"-O0WL.T".)[%*H/\t"0!G))3,pH8"1+eT12(oU#"2/TO\pdFY%"3+)*R4aVWg#)"5+uoI2(p =y`""6sQ#[a.PHcxG"7+jJ!4bXn$44a"9gvwUrt]8a%4-"Fd^bm<,FdoF^uF[ho,&*&,2FignF+F]q" '_j$)F\q"(%eBnF`q")CesFFcq")!GDU'F[as")o.T%)F``s")c-seF]^tFdielF+FeqF+F+*&,2F]q !'[[[!&?.%F+F\q!(STR#F`q!(K%>jFcq!(DSm*F[as!(x$)e)F``s!(:x6%F]^t!'VN#)F+FfuF+F+ /FbpFa_bm/FfpFb_bm/Fi`oFc_bm/F_bpFd_bm/FiapFe_bm/FioFe`qF^r,$**F[vF\v-F-6#,&Fje m"$V$FerF+F+F]qF]v-Fc[l6#,$*&,&-F_jl6#FdielF;-F_jl6#"'VN#)F+F+F]qF+F;F+F\vQ:Pla nar~trees~with~arity~7F$Q,EIS~A007556F$Fia\m-F_jl6#,$*$,&FboF+FfglF+F;F;7#-F>6, Fav7%FB<$F_eyFifalFgn77FinF+F5F2F^blFihy"%#3""%m$*"&'e%*"(q;4"")AX<9"*E^\/#"+Y^ EXK",!>NJ=c".inp;P0""/')3%fo#G@"01d&>Qm0Y"25F'RO48j5"3-=FeS`T2E"4Y1p8lKp5x'"6mA #olQY<.c=F^cbmF^rF^r*&F]qF;)FjdnFfjlF+FavQ,EIS~A000629F$Ffbq,(F`\alF+*&F*FdpFfg lF+FboF[alF+7#-F>6,F^hl7%FB<$F_eyFh\alFgn77FinFinFinFinFinFinFhgnFeen"&?N#"&?2* "'![7$"'?z**"(?v,$"(+;v)")GDbC")gp-n"*?f#)y""*+G"yY"+!3%4.7"+%QC![I"+?NH?wF[ebm <*F_u,(F_^xF+*&,(F\qF^vF]qF4Fd]mF+F+FfuF+F+*&,(F2F+F]qF\cuF\qF+F+FjzF+F+F[pF^pF apFfavFepFgavF^rF[^alFavQ,EIS~A052793F$"$](,$-Feix6#,$FafmF;F;7#-F>6,F^hl7%FB<% F^gmF\[`l/FbsFajxFgn77FinF+F4FabnFj]uF\ehl"&%GD"'CiX"(wFq*"*))=rP#"+gD.$f'"-#*z 27U?".+%GD%p)p"0Ki+%R+I\!R%=4s2Z?FhfbmF^rF^r,$*.F[vF\vF5FfrFg[yFfr* (,&F+F+*&,&-Fc[l6#Fh[yF+F;F+F+FhhbmF;F;F+,&FhhbmF;F+F+F+,&F;F+FfhbmF+F;FfrF]qF] v-Fc[l6#,$*&,&-F_jl6#FghbmF+Fh[yF;F+F]qF+F;F+FfrFavQ,EIS~A052807F$"$p(Fcio7%-F> 6,Q/Set~partitionsF$7%FB<#/FB-FF6#FhhlFgn77F+F+F5FhpFcdmFdinFabjlFfeo"%ST"&Z6#" 'vf6"'q&y'"((f8U")PWkF"*A$**3>"+X&eHQ"",Z@9![5",/)p['G)"-fh!o2#o".d]?UF$e"/s`Be Ts^FcioF^rF^rF^rQYSet~partitions~with~all~blocks~of~size~>=1,~Bell~numbersF$Q2B ell,~EIS~A000110F$Fcdm-F>6,Q%mapsF$FiibmF^jbmFcioF^rF^rF^rQinStrictly~increasin g~partial~maps~(Borwein~et~al),~Bell~numbersF$Q;Borwein~et~al,~EIS~A000110F$Fcb n-F>6,Q/Balls~and~UrnsF$7%FB<$/FB-FF6#%"UG/F[\cmFhhlFgnF^jbmFcioF^rF^rF^rQjoWay s~to~place~balls~into~urns:~distinguishable~balls~and~indistinguishable~urnsF$Q :Bell~numbers,~EIS~A000110F$F]`n,(FhdyF+FidyFboFjxF+7#-F>6,F^hl7%FB<$F_eyFc_`lF gn77FinFinFinFinFcelFcen"$?%"%g7"%sM"%s!*"&gG#"&+h&"'/\8"'w">$"'3^u"(+*><"(!oJR "(_B"*)")/M0?")Sf#[%")gRh**F_\cm<(,(F_^xF+*&,(F2F+F\qF^vF]qF^vF+FfuF+F+F\a`lF+F _uF[pF^pFfavF]glF^r,$*0F[vF\vFh[yF5F5FfrFcgyF+F]qFb^nFa^alF+FigyF+FbrFavQ,EIS~A 052760F$"$<(,$*$,(F;F+F9F5F[alF+F;F;7#-F>6,Ffan7%FB<#/FB-Fe^l6#-FL6%FNFNFcjblFi s77F+FinFinF5F+FinFdpFdpF+FiqFjqF2FabnF[tFcelF[amFd]qFijpF\dqF^f[lF]c`lF]^cm<', (FeqF+FfuF5FjblF;FdoF[pF^pFac[l-F^cl6$,$*&,*F[yF+Fccl!#EFfclF4FdclF2F+FhclF+#F; F[_q/Fccl-F-6#,(F;F+Fj]oF5F_dlF+,$*&,&*$-F-6$Fg_cm$"+$\Fn;(FgdlF4F1*$F\`cmFdpF[ yF;),&*$F\`cmF5F5F[`cmF+F]qF+F;FavQ,EIS~A052922F$"$2*,$*&,(F*F^vFjxF4F;F+F+Ff]l FboF;7#-F>6,F`el7%FB<#/FB-FL6$Fd^l-FV6&FNFNFNFd^lFgn77F+FhpFjqFc\lF]wFdeyFhi_m" &Sa&"'SQ["(Sur%Fe_fl"*+?v)f"++c-kw"-+ONfe5".+;C4#p:"/+?*H"e%[#"0++w(zb%=%"1+g,! *fVpu"3+g,E:A_39"4+g8.%4t$yz#"5++ORi>['*QeFf`cm<&FdoFcjwF[gx,&*&,(F\qF;F]qF1F\c uF+F+FeqF+F+*&F\clF+FfuF+F+*&F\clF+F_glF+F]qFavQ,EIS~A052644F$"$!fF\ebm7#-F>6,F ^hl7%FB<$Ff\alFddilFgn77FinFinFinFinFcelFcenFa_mFdgmFif[l"%CIFeen"%?z"&!)=""&gr ""&CS#"&gF$"&!oV"&?r&"&SM("&CI*"'!G;"F\ebm<(F_uF[pF^pFfavF]gl,&FbhnF+Fji_lF+F^r ,$*0F[vF\vF5Ffr-Fc[l6#F[yF+F_^alF+F]qF`^alFec`mF+FigyF+FfrFavQ,EIS~A052762F$"$> (,$*(F*F+,&Fd[wF+F*FdpF+Ff]lFboF;7#-F>6,Fj]l7%FB<#/FB-FV6$-FL6%Fi^lFd^lFd^lFain Fis77FinFgimFd]mF`crF[tF]`oF\`nFejrFihpF__lFe[qFd]qFfinFbblFccqF``nFcgqFdiqFb[r Fa]rFe^mFbdcm<&F_u,(FeqF+F[yF+F]qFa`lFf[[lFbjz,&FdpF+F]qFe]l,$F]qFe]lQ)~7~n~+~4 F$Q,EIS~A017029F$"$`**&Ff]lFboFfglF+7#-F>6,Fav7%FB<%Fi[nF^d`m/FB-FL6%FbsF_sF_sF gn77F+F4FgimFcirFd\v"%J;"&V<""&,f*"'4o()"("px))")65k)*"+Lib$>"",ThtCc""->tO[+A" .rsI_wJ$"/T637%>L&"0t9&Gj%)*4*"2J(z3V(fOk""3z&>"f)4AE8$"4,gn&G`wk#G'"6@+gh\QmiE K"Fgecm<%FdoFd\x,(*&FhuF+FeqF+F+*&F\glF+FfuF+F+FjzF+F^r*&Fg[yF+F]qF+FavQ,EIS~A0 01339F$FhddlFbg_l7#-F>6,Q?Ways~of~parenthezing~a~productF$7%FB<#/FB-FV6&FdipFaa _lFdiclFd\[lFis77FinFinF+F5FhpFb\lFc\lFd\lFe\lFf\lFg\lFe[dlFf[dlFg[dlFh[dlFi[dl Fj[dlF[\dlF\\dlF]\dlF^\dlFbg_l<&F_uF[pFgbl,&*&,&FboF+F]qFdpF+FeqF+F+FduF+F^r,$F [h_lF`[lQ^pNumber~of~ways~of~completely~parenthesizing~a~product~of~n~letters,~ Catalan~numbersF$Q^sEIS~A000108,~http://www.astro.virginia.edu/|ireww6n/math/Ca talanNumber.html,http://forum.swarthmore.edu/advanced/robertd/catalan.htmlF$Fei n,$*&Ff]lF+,,F+F+F*F;F9FboF[alF5FjxFboF;F;7#-F>6,Ffan7%FB<#/FB-Fe^l6#-FL6%Fg[gl Fc`mFNFis77F+FinF5FdpF2Fd]mFj`mFe[qF\f[lF_^vFdhx"%M;"%5N"%-w"&uj""&I`$"&ii("'mW ;"'-\N"')pl("(5>l"Fhhcm<'FdoFdflF[pF``o,,FeqF5FfuFboFjzFboF[rF;FjblF+-F^cl6$,$* &,*!$,"F+Fccl!$p%FfclF_ejlFdclFi`mF+FhclF+#F;"%^;/Fccl-F-6#,,F+F+F0F;Fj]oFboF_d lF5FjemFbo,$*(,&F;F+-F-6$F^[dm$"+$[v\j%FgdlF+F+,*Fb[dmF+*$Fb[dmF5Fdp*$Fb[dmF4F2 *$Fb[dmFdpF7F;),*F+F+Fb[dmF5Fg[dmF5Fh[dmFboF]qF+F;FavQ,EIS~A052930F$"$;*,$*&F*F +F\iilF;F;7#-F>6,F`el7%FB<#/FB-FL6$FNFeiilFgn77FinF+FdpFi`mF]_m"%![$F\^bl"'g<&) ")g0X;"*+oVd$"++kGH')"-+)[B;H#".+gnVUq"3++?>O>H]D"4+g"o a3!>6&)*"6+S99]&p)fI/%"8++)[$e*p%zXpv""9+!)oilXC&\?"f!)";++#>wzm$3+wG"*QF^\dm<% F^uF_u,(F\^ilF+F_cdlF+FjzF+*&-F^cl6$,$Fi`blFg[lFjjilF+F_glF+,$*(,&Fc^[lF+F;F+F+ ,(F5F+Fc^[lFbo*$F`^dmF5FboF;)F`^dmFfjlF+F;FavQ,EIS~A052631F$"$x&Fgggl7#-F>6,Fdh gl7%FB<&/FdjoFNFjhglF`iglFdiglFgn77FinF+F4FcdmFadq"$Z*"&s/""'Pt8"(0c3#")su,O"*] yS(p",+pip\""-;ngxGN".=!3:>k!*"0+8?nX-_#"11+l'\O?a("3Q4(**R!)=sT#"4G"fcf$Qf6E)" 6!)32V+t%Q;**H"8sGV?'HvKXp_6"9/h,#>)*oPqPen%FggglF^rF^r*&F]qF;-Fc[l6#,$*&,(-F_j l6#Fi[hlF+F;F+Fh[yF+F+F]qF+F;F+Q4Mobiles~of~height~3F$Q,EIS~A000268F$F][^m*(Fdc mF+Ff]lF+,*F+F+F*FboF9F;F[alF+F;7#-F>6,Ffan7%FB<#/FB-Fe^l6#-FL6%Fd^lF^dmFNFis77 F+F+F5FdpF:F\_lF[amFf^qFg^mFi^jl"$2)"%4<"%>O"%kw"&Ii""&qV$"&&ys"'OT:"'7kK"'R7p" (HQY"Fg`dm<(FdoF^uFdflFiir,*FeqF+FfuF;F[rFboFjblF+F``o-F^cl6$,$*&,*FcclFe[qFfcl !#`FdclFj`q!$o"F+F+FhclF+#F;F]ail/Fccl-F-6#,*F+F+F0FboFj]oF;F_dlF+**,&F;F+*$-F- 6$F\cdm$"+<&H@s%FgdlF4F+F+,&F;F+F`cdmF+F+,(F`cdmF5F_cdmF4*$F`cdmFdpF[yF;),(F5F+ *$F`cdmF5F+F_cdmF;F]qF+FavQ,EIS~A052908F$"$))),&FfrF+FiglF\v7#-F>6,F^hl7%FB<%Fc `y/F_s-Fe^lFbjxFghlFgn77FinF+F4F\_lF^fm"%6K"&$Qk"(f7e"")F*[f%"+rH9Afg,&yK""8"pFkF@D 70/?#*":Be3eL-!>n(f`"o"(GgsUx)[J)*zBM)GWF\ddmF^rF^r,$F]jlF`[l FavQ,EIS~A052886F$"$f),$*&FbanF+,*F+F+FjxF^vF[alF5F*F;F;F;7#-F>6,Ffan7%FB<#/FB- Fe^l6#-FL6$-FV6$-Fe^l6#F_]yFNFNFis77F+F+F5FhpF:Fg`mFedmFjhmF[a\m"$x%F]]il"%qA"% d\"&83""&-O#"&,:&"'$R7""'q_C"'X_`"(`!o6"(-!\DFhedm<'FdoF^uFdfl,*FeqF5FjzF^vF[rF ;FjblF+/FioFhp-F^cl6$,$*&,*Fccl!$!>Ffcl!#yFdclFcgqFcdmF+F+FhclF+#F;"$(R/Fccl-F- 6#,*F+F+FjemF^vF_dlF5F0F;,$*(,&F;F+*$-F-6$Fdhdm$"+([#Q#e%FgdlF5F5F+,(FhhdmF2Fih dmF+*$FihdmFdpF7F;),(F+F+FihdmF4*$FihdmF4FboF]qF+F;FavQ,EIS~A052962F$F]fx,$Fcad lF^v7#-F>6,Q2Constant~sequenceF$7%FB<#/FBFe^zFis77FinF4F4F4F4F4F4F4F4F4F4F4F4F4 F4F4F4F4F4F4F4Fcidm<%F_uFd\x,&FeqF;F4F+F4F4Q(All~3'sF$Q,EIS~A010701F$"%65-Fc[l6 #,$F\[oFbjy7#-F>6,Q?Cycles~set~without~fixed~pointF$7%FB<#/FB-FF6#-FV6#-FI6$FN/ FSFhpFgn77F+FinFinFinFinFcelFinFinFinFinFhdilFinFinFinFin"+CecVi"Fajdm<(FdoF[pF^pFapFfav,&FfflF+Fe\oF+F^r,$*.-Fc[l6#,$F]qFbjyF+F5FfrF[vF\ vFhpFfrF]qF\v)F]q,$F]qFh]nF+F[^flQfnpermutations~Sigma~without~fixed~point~such ~that~Sigma^5=IdF$Q,EIS~A052504F$Fhjp,$*$,(F;F+FjxF+F\[oF+F;F;7#-F>6,Fav7%FB<#/ FB-Fe^l6#-FL6$F`_flFNFis77F+FinF+FinF+F+F+F5F+F4F5FdpFdpFhpFe]lFe]lFgimFgimFcuF d]mFcdoF`\em<(Fdo,(FeqF+F[rF+Fe\oF;F[pFfavFgblFhbl-F^cl6$,$*&,,Fccl!$r&Fa]oF\\l FfclFb^mFaeflF+Fdcl!$v$F+FhclF+#F;"%6,F`el7%FB<#/FB-FL6&FNFNFNF`f`mFgn77FinFinFin F2FcelFhyFc_flFignF]bm")g@h6"*+KCK#"++/N4^"-+'4WiA"".+'\YB)=$"/+))=q0F*)"1+Sm0r 6yE"2+![7QZ(*p&)"4+?Vi4T"z8H"6+?bZYz!\'*[5"7+g(pg'>]m1')R"9++/zU'y+mEWf"Fj^em<( F_uFggoF[pF^pF]glFdz,$*&F`hoF+F_glF+F`v,$F`hoF`vFavQ,EIS~A052671F$"$>'-F_jl6#*& Ff]lF+Fh\zF;7#-F>6,F^hl7%FB<%Fj[nFifal/F_s-FLFaf_lFgn77FinFinF5F2F_blFhy"%S?Fee l"'S=B"(+S-$")g8FW"*+C]=(",+GH8G""-+?$33\#".+O^^JC&"0+c!4,C)=""1+!3elHZ)G"2++;+ *fVpu"4+?f5@F1[0#"5+SMX6S*Q\)f"7+?$G*Ry)y.+%=Ff`em<'FdflF_uF[pFdz,(FfayF+*&F`cd lF+FjzF+F+F[rF+F^r*&F]qF;),&F\vF+F_^nFfrFfjlF+FavQ,EIS~A052832F$"$(z-Fc[l6#,(F* F+FjxFfrF9F_em7#-F>6,Fhibm7%FB<#/FB-FF6#-FV6%-FFFhfil-FF6$FNFR-FFF[gilFgn77F+F+ F5FhpFb\lF\`nF\f[lFc]y"%!y#"&WE""&O6'"'wEJ"(#f!o""(!on%*")/Tqb"*'\&=T$"+cM&3<#" ,sQJ9V"",!3]+i(*"-W&y#=uo".w@!p%**)\Fgbem<&FdoF^uFdfl,*F\^ilF+F_cdlF+F[rF5FjzFb oF^r,$*4F[vF\vF5Ffr-Fc[l6##F[yF:F+F4FfrF]qF\v-Fc[l6#,$*&F5Fj_wF]qFj_wFe`blF+-Fc [l6#,$*&F5Fe`blF]qFe`blFfrF+-Fc[l6#,$*&,&F;F+FjdnF+F+F]qF+FfhilF+FdhilF+F_emQeo Set~partitions~with~all~blocks~of~size~<=3,~the~partition~function~G(n,3)~F$Q,E IS~A001680F$Fd]m,$*$,(F;F+F9F+F[alF+F;F;7$-F>6,Ffan7%FB<#/FB-Fe^l6#-FL6%FNFNF[j ^lFis77F+FinFinF+F+FinF+F5F+F+F4F4F5FdpF2FhpF2F]blFgimFgimFcuF\fem<'FdoF[pF^pFi \l,(FeqF+FfuF+FjblF;-F^cl6$,$*&,*FerF+Fccl!#tFfclF4FdclFjqF+FhclF+#F;Fjcl/Fccl- F-6#,(F;F+Fj]oF+F_dlF+,$*&,&*$-F-6$Ffgem$"+M^s">)FgdlF4F^v*$F[hemFdpF[yF;),&*$F [hemF5F+FjgemF+F]qF+F;FavQ,EIS~A017817F$"$%[-F>6,F`el7%FBFcfemFgn77F+FinFinF2Fc elFinFhgnFg`vFignFgen")+k)3"FhenF`^`l",+K33\#Fgdfl".++%=PQlF]bblFj]dlF[^dlF\^dl "5++CE38Kk#*QF\fem<'FdoF[pF^pFdz,(FfflF+*&,*F`qF;F\qFdqF]qFb_cmFjflF+F+FfuF+F+F jblF+*&F\gemF+F_glF+FggemFavQ,EIS~A052697F$"$Y',$*&FafmF;FifbmF+F;7#-F>6,F^hl7% FB<%F]i`mFb`yFdhlFgn77F+F+FdpF\`oFih[m"%!G$"&99&"'id'*")'\v6#"*?vJJ&",S=`@]""-# **eXls%"/'\(R+bQ;"0[+Y&\71i"2[Q_bO#4]D"4S3Z73>4*H6"5W,(=%[G!\0P&"7'*oP64!H9yes# "9W\RCYM-\D`r9":wD$)379u81!=>%)"6,F^hl7%FB<$F_ey/FB-FL6%FNF_sF_sFgn77FinFinFinF 2FcelFidmFg^m"$M%Fadu"%'G#"%+^"&U7""&_X#"&AK&"'gY6"'IdC"'cU_"(yS6""(g#fB"()p!)\ ")?d[5Fd[fm<'F_uF[pF^pFdz,(F_^xF+*&,&F]qF1F\qF^vF+FfuF+F+*&F^cdlF+FjzF+F+F^r,$* 0F[vF\vFccdlF5F5FfrFg[yF+F]qFfrFa^alF+FigyF+F`[lFavQ,EIS~A052749F$"$0(,$*&FieoF +,*F+F+F*FboFjxFboF9F5F;F;7#-F>6,Ffan7%FB<#/FB-Fe^l6#-FL6$-FV6$-FL6$FdfoFNFNFNF is77F+FinF5F5FiqFcuF]hrF\dqFfi[m"$['"%;;"%+S"%O**"&SY#"&_6'"'7<:"'[kP"';S$*"(/v J#"(W,v&")ksE9Ff]fm<&FdoFdflF[p,*FeqF5FfuFboFjzFboF[rF+-F^cl6$,$*&,(F4F+FcclFhh dlFfclF5F+FhclF+#F;F[`n/Fccl-F-6#,*Fj]oF5FjemFboF0FboF+F+,$*(,&F;F+-F-6$F_`fm$" +orJISFgdlF5F+,(*$Fc`fmF5FdpFc`fmF5*$Fc`fmF4F1F;),(F5F+Fc`fmF5Fh`fmFboF]qF+F;Fa vQ,EIS~A052970F$"%U5,$*4,2FboF+F*FhdmF9"%qcFjx!$>)Fe]p"&'RFF[al!&\C#F\[o"&j/&Fj ]p!&i!fF+F[[mF;Ff]lF;FieoF;F]^pF;F^^pF;F_^pF;F`^pF;Fa^pF;F[y7#-F>6,Fg^p7%FB<#/F B-FV6*Fd^lFdfoF]_pFd[mF`_pFd_pFh_pFj_pFis77FiqF_blFchdl"%'H""%s()"&w<'"'kpW"(cu H$")7YoC"*'\%)o="+CvbF9",;+E$)4"",_'***)*\)"-;G$\*4m".%[)\55;&"/w\GCPVS"0#*y:Vf o<$"1O2rNs8-D"2We[oeD[(>"3O.EGDzUh:"4KLiv(*RmkB"F^afm<+/FioF`bfm/F_pFchdl/F\pF_ bl/FeoFiq/F_bpFdbfm/Fi`oFcbfm/FfpFbbfm/FbpFabfm,4FeqF_abmFfu"&k#pFjz!&g)[F[r"&C %=Fjbl!%DSFe\oFecrFb`oFhflFe`oF+FeenF+,2F^]mF+F+F+F`hoF+FgcpF+FhcpF+FicpF+FjcpF +F[dpF+FhcpF^dpQ,EIS~A001555F$"$o$,$*(F*F+Ff]lF;F`adlF+F;7#-F>6,F^hl7%FB<%Fj[nF c`yF^[`lFgn77FinF+FdpFc\nFb`nF]]`l"%w#*"&*G$*"(wt/"")hb(H""*S6sv""+TXG"e#",%=#H k3%"-@u!zM$p"/w3KS&[D""0&=zwO`7C"1'pdQtM#4\"3*[pH>sVR0""4c^)ef&zP.Q#"5")4\9V,a/ Tc"7!3$)41eTb'\*R"Fcdfm<%F^uF_u,(*&,&F\qF+F]qF5F+FeqF+F+F_cdlF+FjzF+F^r,$*,F[vF \vFh[yF+-Fc[l6#FfrF+F]qFg[l-Fc[l6#,$*$F]qFfrF5F+FfrFavQ,EIS~A052852F$"$?),$*&F_ hpF+FhihlF;F;7$-F>6,Ffan7%FB<#/FB-Fe^l6#-FL6$FNF`]dlFis77F+F+F5FdpFe]lFb\lF^blF `blFbblFdblF[ial"$d'"%_7"%&Q#"%WX"%d')"&$\;"&A9$"&k)f"'^S6"''G<#Fhffm<'FdoF^uFd flF``oFd[il-F^cl6$,$*&,*Fccl!$7"FdclFeclFfclFdqFjaclF+F+FhclF+FbgemF[\il,$*(,&F ;F+Fg\ilF+F+Ff\ilF;Fi\ilF+F;FavQ,EIS~A052535F$"$l%-F>6,F`el7%FBF^gfmFgn77F+F+Fd pFcelF`j\lFif[l"&?(=Ff`s"(+/$Q")!G"ol"++g$>D"",+wLDi#"-+K+5(*f"/+!3YW^[""0+G@b" QhR"2+gx.q`?8""3+SyAOdz]M"5+?^KcO5k<6"6+?*4q>&*prKQ"8+?V)pFnMXP(Q""9++/*Rp'[daN 'G&Fhffm<'FdoF^uFjgoFa[q,**&,,F]qF`blFcelF+FcqF+F`qF]blF\qFddmF+FeqF+F+*&,(F]qF ebvFjflF+F\qFboF+FjzF+F+F[glF+FjblF+*&F`hfmF+F_glF+FfhfmFavQ,EIS~A052685F$"$L'* &-F-6#,(F0F;*&F0FdpF*F+F+F+F+F4F*F+7#-F>6,Fav7%FB<$Fbddm/FB-FL6&F_sF_sF_sFNFis7 7FinF+F4FcdmFa_l"$7'"%*Q%"&!*G$"'vWD"(ct,#")k6I;"*VvwL""+L>t66"++ZVQ$*",qeVb"z" -g!\g>w'".&**poz:e"/3h)4')=.&"0!)*e+#HmP%"1W%Qm^4Y#Q"2gSRKz7jN$Ffjfm<(F^uF_uFdh m/FbpFa_l/FioFcdm,&*&,*F7F+F]qFerF\qFgaoF`qF_e_lF+FeqF+F+*&,*F]q!#LF1F+F\qFaqF` qFjaclF+FfuF+F+F^r,$**F[vF\v-F-6#,&FjemF\`oF7F+F+F]qF]v-Fc[l6#*&,&-F_jlFd]\mF+- F_jlFhcrF;F+F]qF+F+#FdqF[tFavQ,EIS~A006632F$F_]v*(FjawF+Ff]lF;F_hpF;7#-F>6,Ffan 7%FB<#/FB-FL6$FcbwFd^lFis77F5F4FhpF2FiqF:FgimFjqFb\lFcdmFabnFd]mFf`mFc\nFcdoFce lF^blF\`oFhjpFi`mF[tFj]gm<%Fd\xFibn,*FeqF+FfuF+F]qF^vF\cuF+,(F``uF`[lF]qFb^nFf^ dlF+,$F]qFb^nFavQ,EIS~A007494F$"%-5,$*(F*F+,&F^fzF+F*FhpF+Ff]lFboF;7#-F>6,Fj]l7 %FB<#/FB-FV6$F`^l-FL6$Fb_pFd^lFis77FinFabnFhjpFj[oFejrFcbnFjdmFh_qFjhmFffqF]jqF a]rFh]]lF\_]lF\\]lF]d^lFd`elFd[[lFgfclF^]blFh_\lFj^gm<&,(FeqF+F\cuF+F]qFhzF_u/F \pFabn/F_pFhjp,&FhpF+F]qFjqFe`lQ*~12~n~+~5F$Q,EIS~A017581F$"$%**,$*$Fh\zF;F;7#- F>6,F`el7%FB<#/FBFii^lFgn77F+F+FdpFd]mFcenFgi_m"%g$*"'Se5"(!)3P"")+%e*>"*+K'HKF [b`m"-+GP2;6".+;%oeZB"/+?jd(yJ&"1+g@,Yn!H""2+g8^ap8M$"3+S1+UJ'4>*"5+!o([OYC$on# "6++[[dE/"HH#)"8++W,:]"QX0jEF``gm<%FdoF^u,(F\^ilF+FdawF+FjzF+*&-F^cl6$,$FeczFbj yF[\_lF+F_glF+,$*&,(FfrF+F_^nF\v*$FdbemF5FboF;FcbemF+F;FavQ,EIS~A005442F$"$%\,$ *$Ff]lFdqF;7#-F>6,F]c_l7%FB<$Fi[n/FB-FL6+F_sF_sF_sF_sF_sF_sF_sF_sF_sFis77F+F:Fe dmF\gpFb]p"%(G"Fgc_l"%Nk"&qG""&5V#"&eP%Fgafl"'qf7"'!\.#"'q(>$"'9.\"'rat"(v:3""( vAc""(v+A#"(0"3JFcbgm<$Fdo,6FeqF_abmF^bvF+F]^tF_blF``s"$Y&F[as"%OXFcq"&\C#F`q"& %GnF\q"'C"="F]qF[jmFignF+,$*2FhuF+F\[lF+F]rF+F\clF+Fg\oF+Fd`oF+Fg`oF+F\cpF+#F+F ign,$F^bvFedgmQOArrangements~of~n~unlabelled~objects~in~9~rowsF$Q,EIS~A000581F$ F[^\m*&Ff]lF;F_^pF;7#-F>6,F^ial7%FB<#/FB-FL6$Fd^lF`_pFis77F+F2F]_lF\[qF]]`m"%1R "&J&>"&cw*"'"G)["(19W#")Jq?7")c^.h"*"yv^I"+1*ye_""+JXRHw",cE(p9Q"-"Gj[t!>"-1kJu O&*".J?er$oZ"/c,"z&=%Q#"0"y]&*G4#>"Fidgm<$Fdo,(FeqF\cuFfuF+F;F+,&Fg[lF+)FhpFhuF `[l,$FicpFajlFavQ,EIS~A003463F$"$u$*&Fc[nF+,(F+F+F*F[yFjxF5F;7#-F>6,Fav7%FB<'/F _\nF[\n/F_sF]\nFbhx/Fe_alFg_al/FBFc_alFis77F+F+F4F]blFj`mFcgqF]\v"%_8"%;Y"&gd"" &3Q&"'7P="'Ksi"(/:9#"(_:J(")+K'\#")'pH_)"*%Q#*4H"*W,^$**"+3e0#R$",WH?"e6Fjfgm<& FdoF^u,(FeqF5FfuF[yFjzF+Fdhm-F^cl6$,$F_cnF`[l/Fccl-F-6#,(F+F+F0F[yFjemF5*(,(Fbo F+Fc^[lFb^n*$,&F+F+Fc^[lF\vF5F+F+,(FdpF+Fc^[lFboF_igmF[yF;)F`igmFfjlF+FavQ,EIS~ A007052F$Fadv,$*&,(F+F+F*F;FjxF+F+Ff]lF^vF;7#-F>6,Fav7%FB<#/FB-FV6$Fd^l-FL6&FNF d^lFd^lFd^lFis77F+F5FdpFe]lFgimFcuFg`mFhjpF[`nF\`nFb[sF__lF`_lFe`qFfdqFjhqFa]rF d`nFe`nFaboF[`lFdigm<&FdoFa[qF]am,*FeqFboF\qF+F5F+F]qF+,(F\qFfrF+F+F]qFfrF_helF avQ,EIS~A000124F$"$'Q*&F_hpF+Fg_`mF;7#-F>6,F`el7%FB<#/FB-Fe^l6#-FL6&FNFNFNFbipF gn77F+FinFinF2FinFcenFhgnFeenFdg[lF]^`lFihemFiablFjablFd[amFg]dlF`i_l"0+!3#QYwM $F]bcm"3+SQgPYm#z""4+SyE^ro3]%"6++Ob1S)>7#>"Fgjgm<&FdoF[pF^pFdb`m*&-F^cl6$,$*&, (F1F+FcclF:FfclFdpF+FhclF+#F+FcdoF[a`mF+F_glF+*(,&F;F+F]gflF+F+F`a`mF;F[gflF+Fa vQ,EIS~A052679F$"$F'-F_jl6#,$*$,&F;F+F]gglF+F;F;7#-F>6,Fdhgl7%FB<)FghglFjhgl/F_ [p-FI6#Fc[pF]iglF`igl/Fc[pFNFdiglFgn77FinF+F2FgdmFg_bl"&WA""'bdC"(v)4e"*+#)>e"" +0D&y(["-vy2b!o"".+lS<-S'"0B3e\"eqE"2.e?>e]-(30*)***Q*RO`WKF c\hmF^rF^r*&F]qF;-Fc[l6#,$*&,(-F_jl6#,&-Fc[l6#*&Fa[hlF+-Fc[l6#,$Fd[hlF;F+F+F;F+ F+F;F+F^_hmF+F+F]qF+F;F+Q4Mobiles~of~height~6F$Q,EIS~A000406F$"$-$,$FdcvF^v7#-F >6,Fj]l7%FB<#/FB-FV6$F[jdlFe^zFis77FinF:FcdmFc\nF\`oFbbnF]`oFedmFfdmFgdmFhdmFab lFj[qFd]qFbcoFj`qFfbqFadqF]fqFhgqFdiqFd_hm<&F_u/F\pF:,(FeqF+F^vF+F]qF1/F_pFcdm, &F4F+F]qF2FbjdlQ)~6~n~+~3F$Q,EIS~A016945F$F]_dm*(F*F5F_hpF;Ff]lF;7#-F>6,F`el7%F B<#/FB-FL6&FNFNFd^lFbipFgn77FinFinF5F2FeclFhyFdelF]g[lFha`mFgablFa[am"*+Se*>Fc[ am",+[7it$Fe[am".+gd?P:*Ff[am"1+!oZU*\XGFh[am"4+!)[zO!f![4"Fj[amFe`hm<&FdflF_uF [p,(*&,*F`qF;F\qF[yF]qF\cuFboF+F+FeqF+F+FdawF+*&FjzF+F]qF+F+*&,(F``uF`[lF]qFfrF g[lF+F+F_glF+Fi[lFavQ,EIS~A052657F$"$/',$*&Ff]lFboFj`lF;F;7#-F>6,Fhcm7%FB<#/FB- FL6%Fd^lFd^lFhalFis77F+F5F4FdpF2FiqF]blFjqFcdmFd]mFc\nFcelFh`mF[tF_blF[amFedmF` blF`]lFihpF]^sF]bhm<&FdoFdhmF]am,0FeqFboFfuFboFjzFboF[rFboF\qF+F]qF:Ff`mF+,*F\q Fbr#FcdmFcuF+F]q#F4Fdp-F^cl6$,$*&,&FfclF+F^vF+F+FhclF+Fe^al/Fccl-F-6#,*F+F+F0F+ FjemF+Fj]oF+F+,$F\qFbrQYnumber~of~ways~to~make~n~cents~with~coins~of~1~1~4~cent sF$Q,EIS~A001972F$F]jn*.Ff]lF;FieoF;F]^pF;F[[mF;F_^pF;F`^pF;7#-F>6,Fdgal7%FB<#/ FB-FL6(Fd^lFdfoF]_pFd[mF`_pFd_pFis77F+Fc\nFee_l"%YE"&FG#"'([z""(_OK""(78K*")tjV j"*tKp?%"+el#\t#",)*)\d]<"-R5D(o5""-z6,Q0Mult iples~of~19F$7%FB<#/FB-FL6%-FV65FNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFd^lFd^lF is77FinF\_lF`rFgdmFeinFbblF[[qF_\rF^]]lFei\lFfhvFd[[lF]bhlF^`lFee_lFe[[mFjadlFj [[lF`_mFgddlF`c`mF[ghm<%F_uFb`l,&FeqF+F]qFd]l,$F]qF\_lFjghmF_ghmQ,EIS~A008601F$ "$J$,$*&Ff]lF+,*F+F+F*FboFjxFboF[alF+F;F;7#-F>6,Ffan7%FB<#/FB-Fe^l6#-FL6$F[^`mF g[glFis77F+F+FdpF]blF\`oF]`nF\jnF]`alF]fr"%DQ"&'H5"&9x#"&*fu"',3?"'/0a"('*[X""( ,i"R")$RT0"")%ou$G")esPw"*$o(e0#F]hhm<'FdoF^uFa[qFe`il,*FeqF+FjzFboF[rFboFjblF+ -F^cl6$,$*&,*FcclFhflFdclF5FfclFg`mFb_\lF+F+FhclF+#F;Fbg[m/Fccl-F-6#,*F+F+F0Fbo FjemFboF_dlF+,$*(,&F;F+-F-6$Fbjhm$"+S(p]r$FgdlF+F+,(*$FfjhmF5FdpFfjhmF5*$FfjhmF dpF[yF;),(F5F+FfjhmF5*$FfjhmF4F;F]qF+F;FavQ,EIS~A052982F$"%b5,$*&-Feix6#,$F*F;F +F*F;F;7#-F>6,Fav7%FB<$F]i`mFciclFgn77F+F+F4FcuF]jqF`bfm"&2o"F\i[l"(pHy%"*++++" "+"pZzN#",CUO<>'".PSRg@z""/'HvB"Rpc"1v$f$o]>Y>"2Oz#z.%fd?("4$z:)4:0BC'G"6w6,F`e l7%FB<#/FB-Fe^l6#-FV6%FNFNFbf`mFgn77F+F4Ff`mFchdl"%%y#"&?v%"'SM(*")SYEB"*+KWN'" ,Sq%e_>"-+c5amm"/+wH%4P]#"1+kSUQyD5"2+;G%Q'=Hb%"4++%Q))="\i<#"6+!G@j^))o_96"7+? *H@S%4DP)3'"9+S#egYm!z1zLN";+SAo_9,N"*3sr@"=+!39+i2%z(\f'z39">++Siib!RuR#)[')>' *F^]im<%FdoFd\x,(F_^xF+*&,&F]qF[yF7F+F+FfuF+F+FjzF+*&-F^cl6$,$FhclF`[lFihgmF+F_ glF+,$*(F5FfrFaigmF;FbigmF+FfrFavQ,EIS~A052590F$"$N&,$*(F*F+,&Fa`lF+F*F5F+Ff]lF boFbo7#-F>6,Fj]l7%FB<#/FB-FV6$-FL6%Fa`pFd^lFd^lFainFis77FinFb\lFcelFj`mF]hrF\\l FgclFe[qF[emF_aqF\dqF[[qFhinFd\rFe^mFd`nFadvFh[]lF[jnF_ealFchdlFh_im<&F_u,(FeqF +F[yF+F]qFgdl/F\pFb\l/F_pFcel,&FdpF+F]qF]bl,$F]qF]blQ*~10~n~+~4F$Q,EIS~A017317F $"$u*,$*&Ff]lFboFb]glF;F;7#-F>6,Fhcm7%FB<#/FB-FL6%Fd^lFd^lF[^glFis77F+F5F4FdpFh pF2Fe]lF:FgimFddoFcdmFabnF\_lFc\nFcelF\`oFi`mFbbnF_blF]`oFc\lF_aim<)FdoF_`oFdhm F]amF``o,6FeqFboFfuFboFjzFboF[rFboFjblFboFe\oFboFb`oFboF\qF+F]qFcdmFb[sF+F[ao,* F\q#F+Fb\lFciamF+F]q#F:Fb\l-F^cl6$,$*&,,FfclF5FdclF4Fa]oF4FdaoF5F^vF+F+FhclF+Fj cglF[dglF+,$F\qF]bimQYnumber~of~ways~to~make~n~cents~with~coins~of~1~1~7~centsF $Q,EIS~A008725F$Ffhv-F^cl6$*(FddpF+FjdpF;-F_jl6#,$*$,&F;F+FaepFdpF;F;F+Fbep7#-F >6,Q7Necklaces~of~4~coloursF$7%FB<'FgfpFhfpFifpFjfp/FB-FI6#-FV6&F`fpFafpFbfpFcf pFis77FinFdpF]blFcelFidmF]jn"$+("%WB"%I#)"&W"H"'o\5"'/8Q"(+&)R""(CA;&")S^<>")WH er"*!fR%o#"+W0e55"+SPw6,QKWords~with~at~most ~3~consecutive~b-lettersF$7%FB<&/%"aGFffp/%"bGFffp/%"XG-Fe^l6$FheimFiedl/FB-FL6 $Fjeim-Fe^l6#-FL6$FfeimFjeimFis77F+F5FdpFiqFcdmFhjpFb[sFc^mF]jn"$,%"$t("%!\""%s G"%Ob"&r1""&p0#"&['R"&Ck("'7t9"'`RG"'PtaF[eim<'FdoFa[qF]am,,FeqF+FfuF+FjzF+F[rF +FjblF;Fd\el-F^cl6$,$*&,*FdclFgclFa`\mF+FcclFdbnFfclF\dqF+FhclF+#F+"$j&/Fccl-F- 6#,,F;F+F0F+FjemF+Fj]oF+F_dlF+,$*(,**$-F-6$F]him$"+P1!z=&FgdlF5Fbo*$FbhimF4F^vF bhimF;*$FbhimFdpF[yF;,*F+F+FbhimF+FfhimF+FahimF+F+)FhhimF]qF+F;QinWords~with~at ~most~3~consecutive~b-letters,~Tetranacci~numbersF$Q,EIS~A000078F$Fgim,$FghnFbo 7#-F>6,Fj]l7%FB<#/FB-FV6$F[jdlF_camFis77FinFiqFb\lFf`mF^blF[tF`rF]hrF`blFb[sF\] sFf^sFe[qFijpF[_qFe`qFabqF\dqFheqFcgqF_iqF\iim<&F_uF^fhl,(FeqF+FboF+F]qF1Fj^z,& F5F+F]qF2FbjdlQ)~6~n~+~2F$Q,EIS~A016933F$"$Y**&Ff]lF;F[[mF;7$-F>6,F^ial7%FB<#/F B-FLFaiflFis77F+FhpFc\nFf^qFh\`lFdafl"%ha"&X=#"&"Q()"'D&\$"(,")R""(0Cf&")@'pB#" )&[y%*)"*TR"zN"+ldlJ9"+hIiEd",XA\1H#",")*ofi"*"-Df(Q]m$".,P]:gY"F\jim<$Fdo,(Feq F[yFfuF+F;F+,&FfhilF+)FdpFhuFj_w,$F^]mF`fhmFavQ,EIS~A002450F$Fb[`l-F>6,F`el7%FB FajimFgn77F+FhpFc\l"$5&"%%=)"'+Q;"(?>$R"*+))45""+?>?BN"-+?jNo7".+)3*GM2&"0+S!>" 4BB#"2+OR]U3:2""3+!)[ZsQ%=d&"5+#>w.tdK-7$"7++_J**oZ&R@(="9+!orSyI2Jp")>"":++_:" >]K88bZ")"<+!oJ-]P!ec%pBme">++gp@5IF7(y+MeW"@++kW<=58DsH1smc$F\jim<%FdoFejflF[g x*&Ff[jmF+F_glF+Fh[jmFavQ,EIS~A052654F$"$,',$*&-Feix6#,&FfglF;F+F+F+FeglF;F;7#- F>6,F^hl7%FB<%F^hx/Fbs-FL6$F_sFBF_eyFgn77F+F+FdpF^blF_[q"%#*H"&1f%"'(GX)")EH==" *YwzZ%",*)3wHC""--]/bSQ"/5Rq3d28"0,q(HzIk["2'H*QVVrK'>"3#y"y-1T.X&)"5B$Rs(*\M1) *)R"7'zsrdQ))zr$*)>"9#=cW3,6?aM]0"":r_k=Ku#yK26If"6,Fav7%FB<#/FB-FL6$Fb]clF]`pFis7 7F+F]blFb^m"$5)"%!H("&5c'"'!\!f"(5WJ&")!pHy%"*5sYI%"+!*[?uQ",5S%y'[$"-!4'f5QJ". 5[O&HCG"/!H$Ge'=a#"05'\X#zwG#"1!\Y4K6*e?"25%=&))=?I&="3!plm*p"=xm""457**pHNY4]" "5!*3#*Hn<<&3N"F[`jm<%FdoFi`n,&FeqF:FfuF;,$)F:F^hoF]bl,$F\dp#F]blF:FavQ,EIS~A00 3952F$"$6$,$*&F*FdpFf]lF;Fbo7#-F>6,F`el7%FB<#/FB-FL6'FNFNFNFd^lFc`mFgn77FinFinF inFinFeclFhyFaiwFg`vFdg[lF]^`lF^^`lF_^`lF`^`lFff_lFa^`lFb^`lFc^`lFd^`lFe^`lFf^` lFcgdlF`bjm<(F_uF[pF^pFfavFahnFegdlFfgdlF5FavQ,EIS~A052683F$Fea_l,$*$Fa^pF;F;7# -F>6,F`el7%FB<#/FBFh_pFgn77F+Fe]lFabqF^fx"&Cw&"(So,#")!G2Z)"+?nl]T"-?jxOCB"/g"3 p^VY""1+7d$=Y]-""2+C)\8c&G*y"4+;_Q`r)**Hm"6+cY0e4$))HLg"8+)Gc$*)QV&Gj7f":+S-T#Q e0(\k#3i"<+!)o%*H)QDqOiD`p">+?(yOw?@gn@\Pu#)"A+?(=NA;s'y68S7dU5"C+g(*)zsd(RjnWP \(>mQ""E++kc=>3mvoaU7\wET>F]cjm<$Fdo,&*&,&F]qFa`lFa`lF+F+FeqF+F+FfuF+*&F[dpF+F_ glF+F[dpFavQ,EIS~A051188F$"$9'**F_hpF;FahpF;F]`\lF;,&F;F+*$F*Fi`mF+F;7#-F>6,Fhc m7%FB<#/FB-FL6&FbipFhipF^a\l-Fe^l6#-FL6@FNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNF NFNFNFNFNFNFNFNFNFNFNFis77F+FinF+FinF+FinF+FinF+FinF+FinF5FinF5FinF5FinF5FinF4F ^ejm6, F``x7%FBFb`xFgn77FinF+F5F:Fgcl"$0&F_dgm"&\i%"'_j_"((\/m")![m/*"+h:dT8",)3#G#R@" -h^"frk$".OD^Fjh'"0ly>jr=F""1c!4u7V9e#"2h0Fnb-V^&"43qS#RekAO7"5`x7"3^#[D,H"6?XP 8..zv:6(Fc^[nF^rF^rF^rQ;Labelled~trees~of~height~3F$Q,EIS~A052513F$Faco,$*&Ff]l F+,*F+F+F*F^vFjxFboF9F5F;F;7$-F>6,Ffan7%FB<#/FB-Fe^l6#-FL6$Fc`mF`gzFis77F+F5Fiq F^blFb^mF^_m"%Y5"%qN"&!>7"&=;%"'%4U""'Q^["(mjl""('=bc")9!3$>")#o@f'"*-22D#"*U%R %o("+mjhBE"+yled*)",#e8JeIF_`[n<&FdoF[hoF]am,*FeqF5FfuFboFjzF^vF[rF+-F^cl6$,$*& ,(FheclF+FcclFb_\lFfclFcuF+FhclF+#F;Fabq/Fccl-F-6#,*F+F+F0F^vFjemFboFj]oF5,$*(, &F;F+-F-6$F\igm$"+)=K*GHFgdlF+F+,(*$F[c[nF5FdpF[c[nF4*$F[c[nF4F1F;),&FdpF+F[c[n FboF]qF+F;FavQ,EIS~A052543F$"$y%-F>6,F`el7%FBFf`[nFgn77F+F5FcuF\[qFdel"&?n$"'?J v")+G*z""*+3]"\",S)RB5:"-+sqIc^"/+%=l:l$>"0+c=k>S$z"2+)o)\3'\@N"4+on&)pmRKo""5+ g(p%o%Q4/i)"7+gPh1*y+2"4Z"9+?Vi`]s([UKt#";+![Wu[gfNCP(z;"=+g*o&e$eh,(=lk*3"">++ [C"RQ2U*\%=dSuF_`[n<&FdoF]am,*FffnF+FhfnF+FbfhlF+F[rF+F[]gl*&F^b[nF+F_glF+Fhb[n FavQ,EIS~A052674F$"$A'*&Ff]lF+,(F;F+F*F+FjxFdpF;7#-F>6,Ffan7%FB<#/FB-Fe^l6#-FL6 %Fd^lFNFf[mFis77F+FinFdpFdpFf`mF_blFcgqFih[m"$C(Fiim"%gY"&;<""&c.$"&?s("'W')>"' Cv]"(+@I""('>KL"('fS&)")!Qp=#")k<.cF_e[n<%FdoF[p,(FeqFdpFfuF+FjzF;-F^cl6$,$F\h^ lF]ebl/Fccl-F-6#,(F;F+F0F+FjemFdp*(,&#FdqFiqF+FfeblFbrF+,(FbrF+FfeblFgr*$,&FgrF +FfeblFbrF5F7F;)Ffg[nFfjlF+FavQ,EIS~A052923F$"$3**.Ff]lF;F_hpF;FeazF;F[`\lF;F]` \lF;F_`\lF;7#-F>6,Fhcm7%FB<#/FB-FL6(Fd^lFbipF^bzF[a\lF^a\lFba\lFis77F+F+F5F5F4F dpFhpF2Fe]lFiqFgimFjqFcdmFcuF\_lFg`mF`crFh`mF]_lFj`mFj[oFjg[n<_pFdoF^uFdflFac[l Fbc[lFcc[lFd^gl/Fg`rFcu/Fj`rF\_lFf_glFg_gl/F\brFj[o/FiarFj`m/Fd^s"%"="/Fi^s"%M7 /F]_s"%68/Fa\q"%-=/Fe\q"%z=/F]\q"%D/F_]sF]\dm/Fc]s"$p*/FggrFee[l/F[grF_ejl/F_grFe_v/ FgfrF]`\m/F`hrF[gal/FdhrFidw/F[hrF^_m/F`[sFh^dl/Fe[s"$q'/F\[sFhf[l/F_jrFc[`l/Fc jr"$I&/FhjrFhax/F[jrFgdhl/Ffir"$^%/FhhrFaeal/F]ir"$%R/Fair"$;%/Ff\s"$G)/Fj\s"$" ))/Fi[sFe]fm/F^\s"$S(/Fb\s"$$z/Fg]s"%A5/F[^s"%v5/F`^s"%G6/FgerFf^m/FgdrFcgq/F[e rFdbn/F[drFh_q/F_drFhaq/F^crFf^s/FccrFj[q/FgcrF^`n/FbbrFfdmFf[[m/FjbrF[\o/F[frF h_l/F_frFcco/FcfrFi\v/FcgrF`eal/FcdrF``n/F_brF]hr/F_erFb`n/FcerFh]]lFdc\m/F`arF `cr/FcarFh`m/FfarF]_lFc\o/Fi`oFhp/FiapFe]l/F^`rFgim,^v!+SG)RV%F+F]q!*aZ*3EFeqFc enFfuFhf[lFjzFif[lFcq!$D%Fe\oFf]elF[rF\g[lFjbl"%+mFb`o"&+i"F\q!(v*RhF`q!&XA(F[a sF;F]as"'++CFaasF^_\nFeasF^_\nFiasF^_\nF]bsF^_\nFabsF^_\nFebsF^_\nFibsF^_\nF]cs "'++7Facs"'+?8Fecs"'+g:Fics"'+S9F]ds"'!)y;Fads"'?.>Feds"'+%z"Fids"'+%4#F]es"'+/ ?Faes"'+QAFees"'+s@Fies"'+MBF]fs"'+#H#Fafs"'+%R#Fefs"'?$Q#Fifs"'+kBF]gs"'!))R#F agsF^_\nFegsF^_\nFigsF^_\nF]hsF^_\nFahsFb_\nFehsFc_\nFihsFa_\nF]is"'+!3"FaisF`_ \nFeisF__\nFiisF\`\nF]jsF^`\nFajsF]`\nFejsFh_\nFijsF[`\nF][tFi_\nFa[tFg_\nFe[tF f_\nFi[tFd_\nF]\tFe_\nFa\tF^_\nFe\tF^_\nFi\tF^_\nF]]tF_`\nFa]tF^_\nFe]tFj_\nFi] tF^_\nF^^t"&?@(Fb^t"&+S)Ff^t"&+1'Fj^t"&!o\F]cpFi`glFjbp"&+1$Fe`o"&+G#Fa_t"&+g*F e_tFc`\nFi_tFd`\nF]`tFa`\nFa`tFb`\nFe`tFg`\nFi`tF``\nF]atFif[lF`atFj^\nFcatF\g[ lFfatF[_\nFiatFf]elF\btFhf[lF_btFcenFfjtFe`\nFijtFf`\nF\[uFi`gl,@F]q#"(nF`$Ffa^ lFcq#Fgim"'++gF\q#"&Ff#F\a\nF`q#"%nL"(++S#F[asF\a^l#"'@Gh"(+++#F+-F^cl6$,$*,,,F cclFbbnFdclFabnFfclFabnF`crF+Fa]oFbbnF+FccuF+FhuF+F\[lF+F]rF+F\a^l/Fccl-F-6#,.F i]oF+F_dlF5Fj]oF5FjemF5F0F5F+F+F+-F^cl6$,$*(,,FcclFj_yFdcl"&jl#Ffcl"&zL&"%.CF+F a]o"&*H\F+F]buF+FhuF+#F+Ffa^lFja\nF+-F^cl6$,$**,,FcclF^h_lFdcl"$N$Ffcl"$z'F_bcl F+Fa]o"$4&F+F[huF+FhuF+F\[lF+#F;Fda\nFja\nF+-F^cl6$,$**,*F+F+FcclF4FdclF+FfclF4 F+F[huF+FhuF+F\[lF+#F;"'++?/Fccl-F-6#,,F_dlF+Fj]oF;FjemF+F0F;F+F+F+-F^cl6$,$*&, ,Fccl!(R-_#Fdcl!(^z8#Ffcl!(be4""'dY@F+Fa]o"&Lw)F+FhclF+#F;")+++SFja\nF+-F^cl6$, $*&,4FcclFboFfclF^vFdclFboFa]oF^vFdaoFboFeaoF;FjduF+FbduF5F^vF+F+FhclF+#F+F\t/F ccl-F-6#,&Fe^^lF+F+F+F+-F^cl6$,$*&,*!%TZF+Fccl"%s>Fdcl"%VNFfcl!%M@F+FhclF+#F+"' ++5Fhc\nF+-F^cl6$,$*(,*!$z"F+FcclFj[ilFdclFgdmFfclFceflF+F]buF+FhuF+F[f\nFhc\nF +-F^cl6$,$*&,joFboF+FcclFboFdclF^vFfclF^vFa]oF[yFdaoF[yFdduF[yFjduF[yFbduF[yFea oF[yF\euF1FhduF1FafuFa`lFbfuFa`lFcfuF7FdfuF7FefuF7FffuF7FhfuF7FifuFdqFjfuFgdlF[ guFdqFj\vFdqF\]vFa`lF^]vF[yF`]vF\cuFb]vF1Fc]vF\cuFe]vF^vFf]vFboFg]vF[yFj]vF;F\^ vF+F]^vF\cuF^^vF7F`^vFgdlFgfuF7Fb^vFgdlFd^vF1F+FhclF+F]i`l/Fccl-F-6#,,Fb^^lF+Fc ^^lF+Fd^^lF+Fe^^lF+F+F+F+,$F[asF\a^lQ\onumber~of~ways~to~make~n~cents~with~coin s~of~1~2~5~10~20~50~centsF$Q,EIS~A001313F$Fbg[m,$*$FieoF;F;7#-F>6,Q&WordsF$7%FB <%/FB-Fe^l6#-FV6$FfeimFheimFeeimFgeimFis77F+F5FdpFiqFcuF[tFgclFgaoF_e_lFhbbl"%C 5Ff^\lFdjx"%#>)"&%Q;F`j[l"&Ob'Fa]\lF\i[lFih[l"(w&[5F_g\n<$Fdo,&FeqF5FfuF;F`hoF` hoQMWords~of~n~letters~in~a~two~letters~alphabetF$Q,EIS~A000079F$F2*&F*F4Ff]lFb o7#-F>6,F`el7%FB<#/FB-FL6'FNFNFNFd^lFd^lFgn77FinFinFinF2FeclFa_mFc_flFiyFc]dlFg el")+/.HFja`m"++g,!z%",+)Gs\o".+W\Rh/"Fh`w"0+?Ve!>HHFjdfl"3+![;HzzV-""4+S9]pqmz 1#"5++_zr9OAzVFfh\n<'F_uF[pF^pFdz,&*&,&F\qF;F+F+F+FeqF+F+*&FhdxF+FfuF+F+*&FhdxF +F_glF+F]qFavQ,EIS~A052571F$"$9&,,FfrF+F*FboFge_lF\vFjxF+Fie_lF+7#-F>6,Ffv7%FB< %Fg]_lF_f_lFddwFgn77FinFinFinFinFcelFbjvFg`vFc]dl"(+Gl'"*gtc2#"++w&[E("-+'4;H#G "/+/Jmz17"0+?:h<gbDrH4*"7++3m$)3Q(4'zc"9++g*)G, t6O)Gx$";++%yU2,eE15hl#"=++gX**[&*=5f=[v>,,FfrF+F*F;Fge_lF\v*&Fbg_lF+F*F+FboFjx F;<(F_uF[pF^pFfavF]gl,(*&,(FhzF+F\qFiqF]qF[yF+FeqF+F+*&,&FboF+F]qF1F+FfuF+F+Fjz F+F^r,$F[h_lFbrFavQ,EIS~A052715F$"$r',$*&Ff]lFboF[`\lF;F;7#-F>6,Fhcm7%FB<#/FB-F L6%Fd^lFd^lF[a\lFis77F+F5F4FdpFhpF2Fe]lFiqF:F]blFjqFb\lFcuFd]mFf`mFg`mFcelF^blF h`mFi`mFbbnF_\]n<,FdoF_`oFdhmF]amF``oFj_dl,6,F`el7%FB<#/FB-FV6$FdipFdf oFgn77F+F5F]blFeclFh[m"%SQ"&!3YF]bm")?>K5"*gXz&="++7*er$",+kg\<)".+Ob!*>'>"/+OR a<,^"1+3-B"H$G9"2+Si!pt)\G%"4+!o**4ef>r8"5+?"*)RvDm?m%"7+?$3O9F&QMy;"8+gh6d9.k1 xP'":++kk%Ge7cE3^DF[^]n<'FdoFggoF]am/FioFecl/F_pF]blFb`emF`hoFavQ,EIS~A052576F$ "$>&*&Ff]lF5Fc[nF;7#-F>6,F`el7%FB<#/FB-Fe^l6#FafwFgn77F+F+F2Fecl"$/&Fj^\nFd_[l" (!3+>")SezR"*?>oP*",+?$)[X#"-+[Wmpq"/+))=Y.@A"0+W(ft;fv"2+Kc/*>iqF"4++s]6QP!)3" "5+?RR"=:Iwb%"7+?:8@v4_WG?"8+gDcM1iIp*e&*":+!3'3VR-GC%)[v%"<++?R^afX?r*o*[#Fj_] n<&FdoF^uFdfnFeg`m*&Fc]nF+F_glF+*(,&FfrF+F_^nF\vF5Fc^nF;Fe^nF+FavQ,EIS~A052567F $F_c\n,$*&FjfclF+FieoF;F;7#-F>6,Fav7%FB<#/FB-FL6$Fb]clFdfoFis77F+F4F2FjqFcelFec lFhaqF]wFh[mFgg`lF^]am"%sI"%Wh"&)G7"&wX#"&_"\"&/$)*"'3m>Fe[\lF_i[lFi]\lFha]n<%F doFch\nFd\x,$F]hoF4,$F`hoFb^nFavQ,EIS~A003945F$Fjadl,$*&F*FdpFieoF;F;7#-F>6,F`e l7%FB<#/FB-FL6'FNFNFNFNFdfoFgn77FinFinFinFinFcelFhyFc_flFignF]bmFd_emFe_emFf_em Fg_emFh_emFi_emFj_emF[`emF\`emF]`emF^`emF_`emF^c]n<(F_uFggoF[pF^pFfavF]glFa`emF c`emFavQ,EIS~A052652F$F_]o*&F^cbmF+F*F+7#-F>6,F^hl7%FB<%F^_hl/FBF_hvFghlFgn77Fi nFinF5F2FcelFb[rFad[l"%uv"&G\("'u7&)")+n"4""*U(>f:"+7:%RX#",)*oW)=U"-gE*Qc'y"/I 9X]d!e""0wT0vH_S$"1-qC$\G'Hy"3!yG8boNO">"4QU;2r9*3a\"6?H"QFIlQ@a8F[d]nF^rF^r*(F jdnF+F]qF;FidbmF+FavQ,EIS~A052862F$"$I),$-Feix6#,$F*FboF\v7#-F>6,F^hl7%FB<$F\\f mF`jxFgn77FinF+FdpF_blFhbbl"&++"Fjf`l"(O&Hv"*caVo#",w0'*>5""-++++?^"/C9zA*fl#"1 C!pJ9"o@:"2w@ohm&*Ga*"4[w%z\A96-l"6+++++++pHy%"8o&4<;dH'=$*yx$":;3x#*Qqj2()f!*= $"<'p4"Q=Vq!e**z6lG">/62&Q+Xk&\;,O'HF"@+++++++++S%p!z([FFde]nF^rF^r,$*.F[vF\vF5 FfrFg[yFfrFh[yFfrF]qF]vFa^alF+F`[lFavQ,EIS~A052746F$"$-(,$*,Ff]lF;F_hpF;FeazF;F [`\lF;FbbclF;F;7#-F>6,Fhcm7%FB<#/FB-FL6'Fd^lFbipF^bzF[a\lF\cclFis77F+F+F5F5F4Fd pFhpF2Fe]lFiqFgimFjqFcdmFcuF\_lFg`mF`crFh`mF]_lFj`mF[amFag]n6,F^hl7%FB<$F`jxF`[gmFgn77FinF+F2Fd]qFigjm"&D1&" (o&*)=")@hw&)"+C9Z'e%"-"[O&HCG"/++++Io>"1\WE&)*y:`""3OrEUQq@;8"5@^Uwbld:Q7"77Xi m)Q%=xVl7"9DcEslt)>g)G'R"";K_&=#)=P!zWjJa;"=,3IXkP)H*>]Zq%4#"?%yIA)ea2&R#G^(*)G #G"A\#z(p.Dt)o5pq*o5MS"C+++++++++g\5+uf$4'F_]^nF^rF^r,$*.F[vF\vF5FfrFg[yFfrFh[y FfrF]qF]vFdgyF+F_emFavQ,EIS~A027471F$"$7(*&FieoF+F[[mF;7$-F>6,Fav7%FB<'/FbsF^__ lFcdwF]__l/Fe_al-FVFfeglFdggmFis77F+F5FiqF[tFgaoFhbblFf^\lF^h\nF`j[lFa]\lFih[lF ciel"(3')Q)")KWbLFeiel"*74(o`"+[O[Z@FgielFhjx"-sM&*Qu8FiielF]_^n<%Fdo,&FeqFdpFf uF;F]am,$F^]mFfrF^`^nFavQ,EIS~A009117F$Fei\l-F>6,F`el7%FB<#/FB-Fe^l6#-FL6$Fc`mF dfoFgn77F+F5FcuF]wFcb]n"&S9'"(gXZ"")!o(GT"+gd?@8",gtSjv%".+WHOD!>"/+O&pf6P)"1+G x`m:=S"3+c='*f9W*3#"5+OR'yv@(3q6"6+gh$=Z0L_?q"8+SU^(>]:\8$\%":+?$o4V8aU(HqwHY&)Q)*>#">+SqkeUHi=&Hv(=n""@++Kw"pSN)*[hB?]P8F]_^n<%FdoF]amFi\m,$F]]mFfr F^`^nFavQ,EIS~A051711F$"$Z',$*&Ff]lF+,(F+F+F*F[yFjxF+F;F;7$-F>6,Fav7%FB<#/FB-Fe ^l6#-FV6$FNF_camFis77F+F4FgimFj[oFf\v"$r&"%J@"%`z"&"oH"'r26"'.MT"(TGa""(hzv&"). !*[@")^!)>!)"*,KI*H"+`Z,<6"+6evoT","\3!eb"",`"yK1e"-@T5&p;#F^b^n<%Fdo,(FeqF+Ffu F[yFjzF+Fd\x-F^cl6$,$Fbe^lF_em/Fccl-F-6#,(F+F+F0F[yFjemF+,$*(,&F+F+*$F4FfrF;F+, (FiqF+Fhd^nF[y*$,&F5F+Fhd^nF;F5FboF;)F[e^nFfjlF+F;FavQ,EIS~A001835F$F]cy-F>6,F` el7%FB<#/FB-Fe^l6#-FV6$FNFfevFgn77F+F4Fg`mFedgl"%sO"&?&o"(?V`"")?J3S"+?zt'>"",! [!e'>S".+k!o:+:"/+)GcF&eh"1+wtJD2eF"3+CE_'oC"Q8"4+7X]P/H:*p"6+!o>03U7"R"R"8+Sm< _")y\1rL#":+g&emW[xKSx#["";+![kLf8FwW=3'**"=+gHZQVEw)4I8J1("?++WL*f")*p-yR&z>F& F^b^n<%FdoFd\x,(Ffg`mF+F^_imF+FjzF+*&F^d^nF+F_glF+Fed^nFavQ,EIS~A052677F$Fbecl- F^cl6$*(FddpF+FjdpF;-F_jl6#,$*$,&F+F+FggxF+F;F5F+Fbep7#-F>6,F^hl7%FB<%FffalFifa l/FbsFbhlFis77FinF+F5FdpF]blF^blFijpFedglFf`jm"%/F"%_#*"&m?$"'?F6"'C+S"(gGV""(/ 1<&")qTy=")yajo"*'\)3_#"*A&Q,$*"+gy;YMF]g^nF^rF^rF^rFavQ,EIS~A003239F$"$h(*$Feb xF57#-F>6,F^hl7%FB<%Fj[nFfh_lF^_hlFgn77FinFinF5Fd]mF\f[l"%S<"&[4#"'%y(G"(7nY%") !oRv("+s=d"\"",SiYK:$"-!G")=GF("/?2B&\%=="0?*\7PI,\"2gt/#*QNqT""3+cm_8JCvV"5+g( *eh&RzqV""6+cyx]P3?Q+&"8+kcY;,!\%=7%="9+k!3o?#3[eYRrFjh^n<'FdflF_^dlF_uF[p,,*&, *F`qFcuF\qFf`mF]qFiqFcqFdpF+FeqF+F+*&,*FbqF+F\qFaqF`qFhzF]qF^hdmF+FfuF+F+*&,(F] qFdinF\qFddoFdinF+F+FjzF+F+*&,&F]qF1FhhdlF+F+F[rF+F+FjblF+F^r,$*(F^[nF+F]qF;F`h oF+F5FavQ,EIS~A052883F$"$c),$*(F*F+,&FdqF+F*F5F+Ff]lFboF;7#-F>6,Fj]l7%FB<#/FB-F V6$F]ecmF_camFis77FinF:FcuFcdoFi`mF[`nF]hrFfdmF[\sFcbnFa^mF`_lF[_qFj`qF_`nF[eqF [[qFjhqFgaoF\\oFi^rF_[_n<&,(FeqF+FboF+F]qFa`lF_uF[]glF_`hm,&F5F+F]qFe]lFcecmQ)~ 7~n~+~2F$Q,EIS~A017005F$"$^*,$*&,&F;F+*$,&F+F+FjxFboFfrF+F+F*F;F;7#-F>6,Q@Label led~Non~Plane~Binary~TreesF$7%FB<#/FB-FV6$FN-FL6$FN-FF6$FBFRFgn77FinF+FinF4FinF ihpFin"%]JFin"'?vJFin")+3R_Fin",+/BVG"Fin".+SP_FQ%Fin"1++)GuZo)>Fin"4+?V5xF`^:" Fin,$*&F*F;,&F5F+Fd\_nFboF+Ffr<'F^uFhoF_uF^p,&*&,(F`qFboF\qF1F]qF[yF+FeqF+F+Fiz F+F^r*(,&*$F[vF\vF+*&Fb[lF+F[vF\vF;F+F]qF]v)F5Fi[lF+Fi\_nQ,EIS~A036770F$F\`n,$F ghnF^v7#-F>6,Fj]l7%FB<#/FB-FV6$Fc_nFe^zFis77FinFjqFc\nFi`mF]`oFeclFgdmF]^sFj[qF [emFj`qFccqF]fqFcenFdbnFf]rFg^nF\[qF\gpFh[]lF][qFh^_n<&F_u,(FeqF+F^vF+F]qFdqFdd v/F_pFc\n,&F4F+F]qF:F[anQ)~9~n~+~3F$Q,EIS~A017197F$"$l**$FafmFhp7#-F>6,F^hl7%FB <$F^gm/FB-FL6'F_sF_sF_sF_sF_sFgn77FinFinFinFinFinFcenF^e`l"&+5#"'+_B"(!)Qp#")+! >B$"*+;.5%"++w[>b",?.#[')y".?2XC\>""/+;o#fm">"0+?Z,w"[K"1!)3,36#R!e"3SMH,())f64 ""4![!e5PZ&3a@"5++OF9[uhcWFi__n<)F_uF[pF^pFapFfavFcfy,.*&,.F;F+F]qF\cuF\qFgdlF` qFgdlFcqF\cuF[asF;F+FfuF+F+*&,,F`qFi`mFcqFhpF\qFidmF]qFj[qF]_lF+F+FjzF+F+*&,*Fh bvF+F]q!$D"F\q!#gF`qFgdlF+F[rF+F+*&,(F]qF`blF\qF]blFcbnF+F+FjblF+F+*&,&FhhdlF+F ]qF\cuF+Fe\oF+F+Fb`oF+F^r,$*&F^[nFdpF]qF;FhpFavQ,EIS~A052767F$F\f[n,$*(FeglF+Fb cbmF;FfglF+F;7#-F>6,Fav7%FB<&F_eyF^d`mFdeglF]__lFgn77FinF+FhpF`crFh]]l"%"3""%l$ *"&&e%*"(p;4"")@X<9"*D^\/#"+X^EXK",*=NJ=c".hnp;P0""/&)3%fo#G@"00d&>Qm0Y"24F'RO4 8j5"3,=FeS`T2E"4X1p8lKp5x'"6lA#olQY<.c="7HiSo()[#fv`N&Fbb_nF^rF^r*&FjdnF;FidbmF +FavQ,EIS~A002050F$Fh]]l*&F*FhpFfglF+7#-F>6,F^hl7%FB<$Ff\al/FB-FL6(FNFNFNFNFNF_ sFgn77FinFinFinFinFinFcenFhgn"%?DF\]alF]g[lFb`sFdacmFhicl"'SW:"'S-C"'g.OFdjv"'g Du"(g"G5"(g`R""(![g=F\d_n<)F_uF[pF^pFapFfavFcfy,&FbhnF+*&,&F]qF+F[yF+F+FfuF+F+F ^r,$*0F[vF\vF5Ffr-Fc[l6#F\cuF+-Fc[lFgpF+F]qFd^nFec`mF+FigyF+FfrFavQ,EIS~A052787 F$Fcial,$*(F*F+,&Fd[wF+F*FhpF+Ff]lFboF;7#-F>6,Fj]l7%FB<#/FB-FV6$F[jdlFe_gmFis77 FinFgimFabnFcdoFhjpFddmFj[oFjhrFejrFacoFcbnFc_sFjdmF]^qFh_qFbblFjhmF[eqFffqFahq F]jqFhe_n<&F_uFf[[l,(FeqF+F\cuF+F]qF1/F_pFabn,&FhpF+F]qF2FbjdlQ)~6~n~+~5F$Q,EIS ~A016969F$"$\**(F*FdpFgfzF+Ff]lF;7#-F>6,F`el7%FB<#/FB-FL6'FNFNFNFNF`gzFgn77FinF inFinFinFcelFhyFhgnFeenFignFgenFjgnFienF[hnF[fnF\hnF]fnF]hnF_fnF^hnFafnF_hnF[g_ n<*F_uF[pF^pFfavFgavF]gl/FfpFhyFahnF_glF+FavQ,EIS~A052663F$"$5',$FcaamF^v7#-F>6 ,Fj]l7%FB<#/FB-FV6$F`^l-FL6$Fe_nFd^lFis77FinFc\nFbbnFedmFgdmFablFd]qFj`qFadqFhg qFdbnFd^rFf\vF\gpFj[vFdfwFi\vF_^\mFf\`lFb\vFe_vFig_n<&,(FeqF+FdqF+F]qFhzF_uFfeh m/F_pFbbn,&F:F+F]qFjqFe`lQ*~12~n~+~9F$Q,EIS~A017629F$"$)**,$*&,.F;F+F*F+FjxF+F9 F+F[alF+F\[oF+F;,,F+F+F*F+FjxF+F9F+F[alF+F+F;7#-F>6,QKWords~with~at~most~4~cons ecutive~b-lettersF$7%FB<&FeeimFgeim/Fjeim-Fe^l6$FheimFegclF]fimFis77F+F5FdpFiqF cuF]_lF[\oFcenF_ejl"$k%"$7*"%$z""%DN"%Ip"&CO""&%yE"&cE&"'>N5"'8N?"''4+%"'olyF\i _n<(FdoFa[qF]amFd\el,.FeqF+FfuF+FjzF+F[rF+FjblF+Fe\oF;/FbpFcu-F^cl6$,$*&,,F\]xF +FcclFa[`lFdclFie[mFfclFh_\lFa]oF]jqF+FhclF+#F+"%)>"/Fccl-F-6#,.F;F+F0F+FjemF+F j]oF+F_dlF+Fi]oF+,$*(,,*$-F-6$Fc[`n$"+;Rg'3&FgdlF5Fbo*$Fh[`nF4F^v*$Fh[`nFhpF\cu *$Fh[`nFdpF[yFh[`nF;F;,,F+F+Fh[`nF+F\\`nF+Fg[`nF+F^\`nF+F+)F_\`nF]qF+F;QinWords ~with~at~most~4~consecutive~b-letters,~Pentanacci~numbersF$Q,EIS~A001591F$Fjq,$ *&FieoF+,*F+F+F*F[yFjxF+F9F5F;F;7#-F>6,Ffan7%FB<#/FB-Fe^l6#-FV6$-FL6$-FVFf^fmFN FNFis77F+F5Fe]lFcelFf^qFc_hmF[]\n"%GQ"&LO""&a&["'FH<"'))eh"(#"(EB"y")6S#y#") %o'4**"*t!QHN"+'e2qD""+.*)*oZ%",!)=rWf"",XM$zycFc\`n<%FdoF]am,*FeqFboFfuF^vFjzF +F+F+-F^cl6$,$*&,(FheclF+FcclFjaclFfclF2F+FhclF+#F;Fb`n/Fccl-F-6#,*F+F+F0F[yFje mF+Fj]oF5,$*(,&F;F+-F-6$Faebl$"+kSw2GFgdlF5F+,(*$Fa_`nF5FboFa_`nFdp*$Fa_`nF4F1F ;),&F4F+Fa_`nF5F]qF+F;FavQ,EIS~A052986F$"%g5,$*(F*F+,&F\cuF+F*F5F+Ff]lFboFbo7#- F>6,Fj]l7%FB<#/FB-FV6$F[jdlFainFis77FinF]blFcuFg`mFh`mFj`mF[amF\`nFdinF[\sFgclF idmFeinF^`nFfinF_aqF_`nFfdqFginFa`nFhinF\``n<&,(FeqF+F[yF+F]qF1F_uFi`nF[]gl,&Fd pF+F]qF2FbjdlQ)~6~n~+~4F$Q,EIS~A016957F$"$[*,.F\imF+FefmFbo*&FafmF+F*F5F;FjxF+F 9F+F[alF`[l7#-F>6,Q0Pairs~of~cyclesF$7%FB<$Ffh_l/F_s-FIF[gamFgn77FinFinFinFinFi nFinFijpFdgm"%'*y"&3m("'_Jz"(?(Q))"*?j41""+gh&*o8",!o:'G*="-?ZLI&z#".+GpNTR%"/+ )o59&Ht"1+Ce)QUMH""2+#*p2(fA3C"3+onHF7u=ZF^a`n<*,(*&,(F\qF;F`qF+F]qFboF+FeqF+F+ *&,(F\qFboF]qF+F4F+F+FfuF+F+*&F^hoF+FjzF+F+F_uF[pF^pFapFfavFep/Fi`oFijpF^rF^a`l QJPairs~of~cycles~of~cardinality~at~least~3F$Q,EIS~A052519F$Ff^q,$*&Ff]lF+,,F+F +F*F;F[alF;F\[oF+F9F;F;F;7$-F>6,Ffan7%FB<#/FB-Fe^l6#-FL6&FNFNFNF`gzFis77F+FinFi nF+F5F+F5FhpFe]lFiqFb\lFcelFj`mFcirF`_lFdiqFbg[mFdcoFe\lFh^[l"$!**F`c`n<(FdoF[p F^pFi\lFjd`l,,FeqF+FfuF;FjzF;FjblF;Fe\oF+-F^cl6$,$*&,,Ff]rF+Fccl"%.?Ffcl!$Y$Fdc l!$W%Fa]oFgimF+FhclF+#F+"%>&)/Fccl-F-6#,,F+F+F0F;F_dlF;Fi]oF+Fj]oF;,$*(,&F;F+-F -6$F^e`n$"+V_&>b'FgdlF+F+,**$Fbe`nFhpF\cuFbe`nF+*$Fbe`nF4F4*$Fbe`nFdpFdpF;),*F+ F+*$Fbe`nF5F+Fhe`nF+Fie`nF;F]qF+F;FavQ,EIS~A052532F$Fdc_l-F>6,F`el7%FBFgc`nFgn7 7F+FinFinF2FeclFcenFaiwFiyFd_fl"(SI!HFe_flF`^`l",+W0'G;"-+#>S70$".+[+&3()o"0+SE ZR%3;"1+ghfx%z!Q"2+g\a,tp")*"4+?Jd(>$=mu#"5++'\yn2ax'z"7++gt[4))Hd3CF`c`n<(FdoF [pF^pFdzFegdl,,*&,.F\qFf\`lF]qFhimFcenF+F[asF+FcqFcdmF`qFf^qF+FeqF+F+*&,,F`qFeb vF\q!#rF]q!$a"F^b]lF+FcqF;F+FfuF+F+*&,*F\qFhzF]q!#ZFja_nF+F`qF;F+FjzF+F+FdeflF+ Fe\oF+*&Fad`nF+F_glF+F_e`nFavQ,EIS~A052651F$Fjd[l,$*&Ff]lF+,,F+F+F*F^vF9F;F[alF +FjxF+F;F;7#-F>6,Ffan7%FB<#/FB-Fe^l6#-FV6$-FL6$Fg[flFNFNFis77F+F5FhpFb\lF`rFb_l Fg\`lFih^n"%o?"%?c"&t_""&1:%"'(z7""'QlI"']I$)"(.RE#"(+C:'")4)>n"")!)yVX"*GB[B"" *8ldN$Fig`n<'FdoF]amFidx/FioFb\l,,FeqF+FfuF;FjzF+F[rF^vFjblF+-F^cl6$,$*&,*!$q#F +FcclFd[wFdclFd^rFfcl!$/"F+FhclF+#F;"%*4"/Fccl-F-6#,,F+F+F0F^vFj]oF;F_dlF+FjemF +,$*(,&F;F+-F-6$Fej`n$"+3rqzOFgdlF+F+,*Fij`nF4*$Fij`nF5Fbo*$Fij`nF4F4*$Fij`nFdp F[yF;),*F4F+Fij`nF;F^[anF+F_[anF;F]qF+F;FavQ,EIS~A052985F$"%f5*&F_hpF;F]^pF;7#- F>6,Ffan7%FB<#/FB-FL6$FbipF]_pFis77F+F4F]blFi`mFa_lF[^vFgffm"%gC"%"Q("&V@#"&Ik' "'!H*>"'ryf"(8Oz""(S3Q&")?D9;")hvU["*$o#GX""*]![eV"+]Ta28"+^CjARFe[an<%FdoFd\x, *FeqF^vFfuFboFjzF+F;F+,&Fgcp#F:Fiq-F^cl6$,$*&,&F4F+FcclF+F+FhclF+#F;FcuF]cwF+,$ FgcpF`]anFavQ,EIS~A033113F$"$1*,$*$FhiblF;F;7#-F>6,F`el7%FB<#/FB-Fe^l6#-FV6&FNF NFNFdipFgn77F+F4Ff`mFh^m"%;E"&+K%"'!3c)")!3#z>"*+/&H_",g0pWb""-+GD1M^"/+SIABl=" 0+OJGKDR("2+Wl$=Q1uJ"4++cm7*3ln9"5+Siib7C)4F("7+?N!)4$>x2B%Q"9++oJIjI&oZt:#";+? V)pj]')yOUDG""<+S=Xw7&yK0WI[!)">++?:T'=$e$*Hu[jJ&Fj]an<%,(F\^ilF+Fa^xF+FjzF+Fdo Fd\x*&-F^cl6$,$*&,&FcclF5F4F+F+FhclF+F^h^lF\\clF+F_glF+,$*&Fc\clF;Ff\clF+F;FavQ ,EIS~A052595F$"$S&*@F[gclF^vFifclF;FjfclF;,(F+F+FjxF;F[alF+F;F_hpF[yF_i_nFboFdc mF^vFf]lF[yFfigmF;,*F[alF+F9F;F*F+F;F+F;,0Fj]pF+F\[oF+F[alF+F9F+FjxF+F*F+F+F+F; ,&F[alF+F+F+F;,(Fj]pF+F9F+F+F+F;,,F[alF+F9F;FjxF+F*F;F+F+F;,8F\`\lF+F\^pF+F^_oF +Fe]pF+Fj]pF+F\[oF+F[alF+F9F+FjxF+F*F+F+F+F;7#-F>6,F_gcl7%FB<#/FB-FF6$F[\n1FSFj qFis77F+F+F5F4FhpFe]lFgimFcdmFg`mFi`mFc\lFb[sFjdmF_`nF_\rFe`nFe_ilFe[[m"$m$"$g% F\gnFd`an<_o/F^`rFc\l/FiapFg`m/F_bpFcdm/F\bpFi`m/FfpFe]l/Ffir"&*H(*/Fair"&n])/F ]ir"&(Gu/Fhhr"&2Z'/Fdhr"&(Hc/F`hr"&\)[/F[hr"&LB%/Fggr"&yl$/Fcgr"&q:$/F_gr"&cr#/ F[gr"&ML#/Fgfr"&y*>/Fcfr"&%3J85_="F[as!/&**[c(pI$GFeas"0+w@_83C%Fias"0+Ojs*p`eF]bs"0+oZ=V9.&Fabs"0+wH-q*Q#)Febs"0+++w$R([(Fib s"0+W"Qc,#o'F]csFienFacsF[eanFecsF_eanFicsF\eanF]dsF`eanFadsFaeanFedsF^eanFids" .+WtxwE"F]es"-+7x)40'Faes".+3-jna%Fees".+'\=suCFies"/+G$RPxJ"F]fs".+'\]OBzFafs" /+o"z9Dz%Fefs"/+G&)R&)GKFifs"/+3YG%>5#F]gs"/+oZyV#*oFags"/+'\EApi*Fegs"0+?Jte%3 8Figs"0+w&[/lTAF]hs"0+%e$3iMt"FfbflF;FahsFhfanFehsFifanFihsFjfanF]isFcfanFaisFg fanFeisFffanFiis"0+K%y*[R()*F]js"0+#f!zke!*)Fajs"0+wHqw"f%*FejsF\fanFijsF[fanF] [tFheanFa[tFgeanFe[tFieanFi[tFeeanF]\tFfeanFa\tF]ganFe\tF[ganFi\t"1+g,uJ485F]]t F^ganFa]t"1+[#4n(z@5Fe]tFjeanFi]tF\ganF]^t!+$=4"e))Fhbfl!'l]GF^^tFafanFb^tFefan Ff^tFbfanFj^tF_fanF]cpF`fanFjbpF]fanFe`oF^fanFa_tFdfanFgbfl!$#z/Fcdr"%,W/F_dr"% bO/F[drFbhr/Fgcr"%.D/Fccr"%j?/F^cr"%'o"/Fjbr"%!Q"/Ffbr"%;6/Fbbr"$0*/F_br"$D(/F\ brF\gn/FiarFgaan/FfarFfaan/FcarFe[[m/F`arFe_il/F]arFe`n/Fj`rF_\r/Fg`rF_`n/Fa`rF b[s/Fg]s"'mOg/Fc]s"'s'R&/F_]s"'p<[/Fj\s"'O'H%/Ff\s"'iDQ/Fb\s"'$HS$/F^\s"''>-$/F i[s"'z!o#/Fe[s"'0uB/F`[s"'0+@/F\[s"'Da=/Fhjr"'SN;/Fcjr"'[R9/F_jr"'gl7/F[jr"'O56 FdoF^uFdflFhoF_`o/Fd`rFjdm/Fi`oFgim,boF]q#"/Zpo"*euN"/++;chH!*F^bv#Feddm".+!oV. BqF``s#"'*p2*"-++kcqiFcq#"*$R+fr"-+?:P.yF\q#",dzQ(z!)"-gT7kF%)F`q#".fI_"H`]"0+# f9CorTF[as#"+F8C^d"0++3IcvG"Ffbfl#F+"2++)o16-7>F]^t#"'`&G#".+?>/DC(Fhbfl#"%,))" 1+gH27%e3#Fgbfl#Fddo"0++ON.Sz&-F^cl6$,$F_[[n#F;Fgb]nFf[\lF+-F^cl6$,$F_cnFch`lFd h`lF+#"2x3/tau.b""2++%ev9O\DF+-F^cl6$,$**,&FffqF+FcclFdpF+F[huF+FhuF+F\[lF+#F+" '%yW*FfemF+-F^cl6$,$*&,&Fa`lF+FcclF[tF+FhclF+#F+"%W>/Fccl-F-6#,(FjemF+F0F;F+F+F +-F^cl6$,$*&,&Fccl"%:P!$%oF+F+FhclF+#F+"'cu9Ff[\lF+-F^cl6$,$*(,&F`crF+FcclF5F+F ]buF+FhuF+#F;F_h\nFf[\lF+-F^cl6$,$*(,(F5F+FcclF5FdclF+F+F]buF+FhuF+#F;"%]iF^czF +-F^cl6$,$*&,6FcclF:FfclFiqFdclFe]lFa]oF2FdaoFhpFdduF4FjduF5FbduF+FeaoFdpF;F+F+ FhclF+#F+"%J8Fg`dlF+-F^cl6$,$*&,*FcclFf^mFdclFd\lFfclFc_sFgdlF+F+FhclF+Fd[^nF^c zF+-F^cl6$,$**F[huF+FhuF+F\[lF+F]rF+F]^bnFfemF+-F^cl6$,$*(,&"%Z9F+Fccl"%n:F+F]b uF+FhuF+#F\cuFj]^nFfemF+-F^cl6$,$*&,&FdaoF+F;F+F+FhclF+#F+F_[qFgaglF+-F^cl6$,$F ][^n#F+"$+&Fhc\nF+-F^cl6$,$*&,&!%xvF+Fccl"&%y_F+FhclF+#F+"(wrT"FfemF+-F^cl6$,$* (Fj`blF+F]buF+FhuF+#F+"%#f#Ff^bnF+-F^cl6$,$Fi`bl#F+FabbmF[dglF+-F^cl6$,$Fhcl#F; F_e_lFaj[lF+F``u#")8<2t"*+))y2(Fd`u#"&.;$")gHfBFh`u#FddoF_dbnF[au#F+"*+W*QNF]au #Fefr")GtYUFaau#"(t8^)"*!GtYU,$FfbflFd\bnQCPartition~of~n~in~at~most~12~partsF$ Q,EIS~A008641F$Fgddl,$*0Ff]lF;FieoF;F]^pF;F[[mF;F_^pF;F`^pF;Fa^pF;F;7#-F>6,Fdga l7%FB<#/FB-FL6)Fd^lFdfoF]_pFd[mF`_pFd_pFh_pFis77F+Fh`mFdc_l"%!)e"&()R'"''RF'"(C ar&")!GH$\"*L8u3%"+/E)=G$",'y/"3d#"-+M[iu>".R[jCH\""/_c/aN96"/[\@d4J#)"0Su'*zBw -'"10t6**>k#Q%"2SX!=&QYx;$"35jr)*H[KyA"4+m,"*zz`=j""5J:C6Hxrvk6F[ebn<)Fdo/F\pFh `m,2Feq!%S]Ffu"%G!)Fjz!%/^F[r"%l;Fjbl!$&HFe\oF\`oFb`oF;F+F+/F_pFdc_l/FioFfebn/F bpFgebn/FfpFhebn,0F[e_lF+F`ho#F7FcdmFgcp#F_[qFcuF^]m#!%C5F:Ficp#"&Dc"FeclFjcp#! %W>FhpF[dp#"'\w6Fhgn,$F[dpF]hbnQWPartitions~of~a~set~of~size~n~into~7~non-empty ~subsetsF$Q,EIS~A000771F$Fidw,$*&F_hpF+,,F+F+F9F;F*F;FjxF;F\[oF+F;F;7#-F>6,Ffan 7%FB<#/FB-Fe^l6#-FL6$-FV6$FbipFdipFNFis77F+F+F+F4FhpFiqFcdmF\`oFjhrF[emFihyFee_ lFgdhl"$U)"%(\""%iE"%NZ"%@%)"&w\""&Nm#"&qt%Fbhbn<(FdoF^uFhoF_`oFgbl,,FeqF+FjzF; F[rF;FjblF;Fe\oF+-F^cl6$,$*&,,!$)zF+Ffcl!$9(Fccl!%()=Fa]oFgc`mFdclFjeglF+FhclF+ #F;"%Jq/Fccl-F-6#,,F+F+Fj]oF;F0F;FjemF;Fi]oF+,$*(,&F;F+*$-F-6$Fijbn$"+;-yAcFgdl F5F+F+,*F][cnF5*$F^[cnF4F4*$F^[cnFhpF\cuF^[cnF+F;),*F+F+F^[cnF+F][cnF+*$F^[cnFd pF;F]qF+F;FavQ,EIS~A052977F$"%\5*&FdcmF+,(F;F+F*F5F9F+F;7#-F>6,Ffan7%FB<#/FB-Fe ^l6#-FL6$F^dmFc`mFis77F+F5FdpFiqFd]mF[amFfinF_eal"$G%"$W*"%#3#"%#f%"&G,""&QB#"& o#\"'k'3""'m'R#"'+'G&"(ke;""(%RrD"()QrcFj[cn<'FdoFa[q,(FeqF+FjzF5F[rF;F]amFd\el -F^cl6$,$*&,(FcclF7FfclF:FjqF+F+FhclF+#F5Faco/Fccl-F-6#,(F;F+F0F5Fj]oF+*(,&F;F+ *$-F-6$F_^cn$"+:l(R`%FgdlF4F+F+,&Fc^cnFboFb^cnF^vF;),&F5F+*$Fc^cnF5F+F]qF+FavQ, EIS~A052910F$"$!*)F`]bl7#-F>6,F^hl7%FB<$Ff\alFh]blFgn77FinFinFinF2Fa^mFc`an"%SK "&5q""&[;)"';uO"(SYd""(!R&\'")g:)f#"*%3G85"*[!*)oQ"+IR$3X""+!GDpN&",?N^9&>",s'[ ADq"-%*ejt-D"-?\r=L))F`]bl<'F_uF[pF^pFdz,&FjfvF+F\j\nF+F^rFb_blFavQ,EIS~A052791 F$"$[(*&FafmF5F*F+7#-F>6,F^hl7%FB<$F^gmF\\fmFgn77FinFinFinF2FcelFheqFhf[l"%OQ"& C#G"'C_B"(!o">#")sYeA"*gt3b#"+S)RT8$",S1S?;%"-+7x#3%f".?br+42*"0+W0U:`Z""1+'ph; Mka#"2+'4?W7%*[Y"3+S'3MA&z]*)Ff`cn<'F_uF[pF^pFdz,(*&,,F`qF+F\qF^vF]qF;FcqF+F5F+ F+FeqF+F+*&,(F`qFboF\qF^vF]qF5F+FfuF+F+Fa]fmF+F^rF^a`lFavQ,EIS~A052745F$F]\bm*( F*F+,(F*F;FjxF5F;F+F+FieoF;7#-F>6,F`el7%FB<#/FB-FL6$FN-FV6$FNFdfoFgn77FinF+F2Fc elF]wFgfoFhfoFifoFj[mFjfoF[goF\goF]goF^goF_goF`goFagoFbgoFcgoFdgoFegoFbbcn<'F^u F_uFdfnFggoFjgoF\hoF_hoFavQ,EIS~A052594F$"$R&,$*&FjfclF+F_^pF;F;7#-F>6,Fav7%FB< #/FB-FL6$Fb]clF`_pFis77F+F2Fi`mFihyFgfbm"%]P"&](="&]P*"'](o%"(]PM#")](=<"")]Pfe "*](oHH"+]P%[Y""+](=UK(",]P4@m$"-](oa5$="-]PMFb"*".](=njxX"/]Pf$=))G#"0](oz"4W9 "Fbccn<%FdoFbivF^az,$F`jyF2,$Ficp#F2FhpFavQ,EIS~A003948F$F]d^mF^r7fn-F>6,QP~Alc ohols~or~Unlabelled~Non~Plane~Ternary~TreesF$7%FB<$/FNFffp/FB-FV6$FN-FL6$FN-FF6 $FBFfnFis77FinF+FinFinF+FinFinF+FinFinF5FinFinFdpFinFinFiqFinFinFabnFinF^rF^rF^ rF^rQPAlcohols~or~unlabelled~~non~plane~ternary~TreesF$Q,EIS~A000598F$F+-F>6,Q> Non~plane~rooted~binary~treesF$7%FB<#/FB-FVF`]_nFis77FinF+F+F+F5F4F2FgimFcdoF\` nFabqF^fmFi[\n"$$)*"%z@"%]["&04""&JY#"&6g&"'7z7"'ZNHF^rF^rF^rF^rQ\sunlabelled~b inary~rooted~trees~with~n~endpoints~inequivalent~under~reflections~in~vertical~ axes.~(Wedderburn-Etherington~numbers)F$Q,EIS~A001190F$F^_l-F>6,Q1Non~plane~tre es~F$7%FB<#/FB-FV6$FN-FF6$FBFa[zFis77FinF+F+F5FhpFjqFbbnFb^mFd\v"$m("%7B"%oq"&l >#"&a*o"'^(=#"'M&*p"(wOD#"()y0t")Vn"Q#")-O-y"*^(QnDF^rF^rF^rF^rQdoUnlabelled~no n~plane~trees~with~degree~>=2~(series-reduced~planted~trees)F$Q,EIS~A000669F$F] hr-F>6,QBUnlabelled~Non~Plane~Binary~TreesF$7%FBF[]_nFis77FinF+FinF+FinF+FinF5F inF4FinF2FinFgimFinFcdoFinF\`nFinFabqFinF^rF^rF^rF^rF`icnFdgcnFedm-F>6,Q6rooted ~identity~treesF$7%FB<#/FB-FL6$FN-F`]wFbjxFis77FinF+F+F+F5F4F2FjqF`crFdinFffqF^ `l"$[&"%E7"%qF"%*H'"&EW""&4K$"&^o("'='y""'[oTF^rF^rF^rF^rFeicnQ,EIS~A004111F$Fb [s-F>6,F``x7%FB<#/FB-FL6$FNFajxFis77FinF+F+F5FdpF:Ff`mFeclFc_lFcc\mFadcm"%U="%m Z"&'[7"&tH$"&6y)"'"QN#"'Z[j"(f6s""(w')o%")Gi#G"F^rF^rF^rF^rQDRooted,~non~plane, ~unlabelled~TreesF$Q,EIS~A000081F$Fgdm-F>6,Fav7%FB<$Ffg_m/FB-FI6$F_sFa[zFis77Fi nFinF+F5FhpFjqF_blF\dqF\hhm"%i5"%>N"&4=""&F.%"'m!R""')p%["(>Gq""(_k-'")[-Y@")]p %o("*y\_w#"*vLU***F^rF^rF^rF^rFavQ,EIS~A032203F$F`_l-F>6,Q0Labelled~mobileF$7%F B<$Fcs/F_s-FIF^hcnFgn77FinF+F+FhpFj[o"$p%"%*)o"'0O7"(p,i#"),\2k"+"3Bcx"",XMu*)\ &".@l$49#)="/*eL&*R_0("1*o(y_OauG"3XIbiB![[E""4*o()GvI"8,92$H luM=)Ri""9Dv>ZNw)\q&[i#*";,-C%*[=dh6ku%e&F^rF^rF^rF^rQQLabelled~mobile~with~cyc les~of~length~at~least~2F$Q,EIS~A032188F$Fh_q-F>6,Q2Unlabelled~mobileF$7%FB<$Fc s/F_s-FI6$FBF[h_mFis77FinF+FinF+F+F5F4F2FjqF`crFejrF[[qFebn"$n&"%%H""%WH"%!z'"& $p:"&)fO"&yb)"'*H,#F^rF^rF^rF^rQSUnlabelled~mobile~with~cycles~of~length~at~lea st~3F$Q,EIS~A052523F$Fa_l-F>6,Fcg_m7%FBFeg_mFis77FinFinFinF+F+F5F4F2F]blFf`mF_b lFc_sFb`nF\_m"$J&"%q5"%Z@"%nV"%&*))"&i#="&)ePF^rF^rF^rF^rQjnNon-planar~unlabell ed~trees~with~neither~unary~nor~binary~nodesF$Q,EIS~A052525F$Fbbl-F>6,Fav7%FB<% FffalFifal/FbsF^i`mFis77FinF+F5FdpF:Ff`mFfdmF]jq"$H$"$i)F]h_m"%6,Fav 7%FB<%F_eglFca_lF^[`lFis77F+F+F5FhpFddoFddmFbblFghel"$F("%L?"%9d"&Oh""&Ld%"'Y+8 "'.3P"(Q)f5"("fNI"(O2r)")Mp.D")M"p?("*,vs2#F^rF^rF^rF^rFavQ,EIS~A000107F$Fdiq-F >6,Fav7%FB<&F^hxF^cx/F_\nF^i`m/F_s-FL6$F_\nFNFis77F+F+F4Fe]lF\_lFjhrFb[rFabbmF` \_n"%:E"%=t"&"\?"&.z&"')*Q;"'*>m%"($**G8"(C'*z$")\S)3"")q6CJ")e\")*)"*UYge#F^rF ^rF^rF^rFavQ,EIS~A001372F$Fe^m-F>6,Fav7%FB<%F]i`m/F_sFd]hl/FbsFaa_lFis77F+F+F4F iqF`crFjdmF[^\mFhjl"%\I"&M3""&2#R"'4O9"'$>K&"(j,*>"(rM](")rg[G"*.&4)3""*SB'yT"+ 71W7;"+U'y([i",wD**4V#F^rF^rF^rF^rFavQ,EIS~A007563F$Fdj\l-F>6,Fav7%FB<%F\[`lF^[ `lFbhxFis77FinF+F5FhpFjqFi`mFe[qFccoFec[n"%N7"%9K"%]%)"&qB#"&w'f"'S,;"'PAV"(OC< ""(q[>$"(U9u)")Xq+C")aY:mF^rF^rF^rF^rFavQ,EIS~A000106F$Fd`p-F>6,Fav7%FBF\f]nFis 77FinF+F5Fe]lF^blF[eq"$e%F^fx"%)\*"&Z\%"')f;#"(_*f5"(1=D&")QsHE"*mn&G8"*&R)Rw'" +/"*zmM",)z3N(y"",')))4IE*"-1Xo#H#[".1]<5;_#F^rF^rF^rF^rFavQ,EIS~A000151F$"$(Q- F>6,Fav7%FB<%Fffal/Fbs-FVFba_l/FBFagmFis77FinFinF+F+F5F4F2F]blF\_lFddmF__lFd_lF ^\\m"$#["$_*"%&)="%lP"%Yv"&@_""&-3$"&?E'F^rF^rF^rF^rFavQ,EIS~A001678F$"$/%-F>6, Fav7%FB<$F]i`m/F_sFeyFis77F+F+F4F]blF]`oFjinF`g]n"%xJ"&I[""&y1("'gGM"('['o""("o $R)")[r=U"*-)GQ@"+w5r"4""+UzH4c",*Q3F)*G"-%fGn\]""-X^c_\y".L#e;\5TF^rF^rF^rF^rF avQ,EIS~A005750F$Fg\cn-F>6,Fav7%FB<%/F_sF[jcnFddw/FBFgfalFis77FinF+F5FhpFd]mF]^ sFee_l"%66"%#z%"&C6#"&))[*"':CV"(G[*>"(7nH*")AnqV"*)RIq?"*c/8()*"+N9'Rt%",M5C>G #"-d[k+06"-c&QdHP&F^rF^rF^rF^rFavQ,EIS~A005753F$Fe\l-F>6,Fav7%FB<%Fad]nF^]w/Fbs -FLFceglFis77FinF+F+F5Fe]lFcelFbbl"$)Q"%];"%$=("&l>$"'-X9"'TAm"(U*oI")y'eV"")t* Hx'"*h%f2d$"-%G17At"F^rF^rF^rF^rFavQ,EIS~A005754F$"$I%- F>6,Fav7%FBFg]^nFis77FinF+F4FcdmF^`nFb]p"%WJ"&v3#"'tF9"(J,+""(7o8(")JAq^"*BYBz$ "+]\(3"G",dv/?5#"-@"H))Re"".I7?<;?"".B5]g!p"*"/zQl1%H.("0@QJk$\>a"1uGE1*pN>%F^r F^rF^rF^rFavQ,EIS~A006964F$"$R%-F>6,F^hl7%FB<$F]i`m/F_s-FL6&FBFBFBFNFis77F+F+Fd pF\_lF[eqF]b^n"%nT"&Zy#"'Z<>"(V(\8"(;8n*")0rHq"*d"zq^"+)[,<%Q",gja(yG"-([ntJ<#" .fm7W6l""/96%4d;E""/>S]R9*o*"09heSEXZ("1")[89:Y*y&F^rF^rF^rF^rFavQ,EIS~A052751F $"$2(-F>6,F^hl7%FB<$/FB-F`]wFchlF^_enFis77F+F+F4FcdmF`_l"$m%Fhfim"&4%="'(>@""'" \:)"(97e&")^wtQ"*y@,s#"+y'R*G>",9')\%z8",&H-5P**"-meW6/s".Y>9%>__"/)[pW$G[Q"0`s gDQA$G"1.dWt*HG4#F^rF^rF^rF^rFavQ,EIS~A052755F$"$6(-F>6,F^hl7%FB<$F\[enF`[gmFis 77FinF+F4FjqFgcl"$j$"%9A"&VS""&d=*"'wYh"(a#*=%"):\(*G"*Q4(G?"++[4N9",T-"",s6 +RO("-`ua/I`".:WL"Q!)Q"/XJ^ccRG"0J"*3#Q[(3#"1Tpz@c$4a"F^rF^rF^rF^rFavQ,EIS~A052 756F$"$8(-F>6,F^hl7%FBF\jxFis77FinF+FdpF^blFcco"%*\""&GE""'k56"(So+""(hdM*")!er $))"*CMF[)"++d2W#)","G,*f4)"-/3PP@!)".#)="GU3!)"/%[>6%*)[!)"0XP@IJq8)"1oMh]o')o #)"2#=i=J4wT%)"3%y\k=8*3a')F^rF^rF^rF^rFavQ,EIS~A052763F$Fhgn-F>6,F^hl7%FB<$F\[ enF]jxFis77FinF+FdpFg`mF\[q"%$>""%[(*"&;H)"')3F("(%3Cl")K2if"*E1(Hb"+3e$>>&",]I !HD\"-_jGe8Z".%*R*4JXX"/_g3;67W"0cr>yzxI%"1Gx.7UjFU"2GRkJv#4oT"3?Q(RLL#QETF^rF^ rF^rF^rFavQ,EIS~A052772F$Fjbgl-F>6,F^hl7%FB<$F]i`m/F_s-FL6'FNFBFBFBFBFis77F+F+F hpF]_lF\ijl"%K="&$e:"'"zP""(-_D"")(p$p6"*p^!46"+?5=n5",n:'GR5"-hFMRA5".*p@U[95" /#y^pfT,""0!RxhGW?5"1F]`">_E.""2ul!))=,K]5"3OBt)4oQJ2""4d+\"[&oA45"F^rF^rF^rF^r FavQ,EIS~A052773F$Fdb]l-F>6,F^hl7%FB<$Fg`enFafenFis77F+F+FdpF^blF[jn"%V9"&))="" 'f=5"'Hv*)"(.^3)")cO6u"*\[8*o"+GV2%['",I*z3ih"-wGCG1f".LX$z-.d"/`"e>oBa&"0)\,s: *oT&"1ET-o$*)4K&"2ws#R"Rf.D&"33He<#\k;?&F^rF^rF^rF^rFavQ,EIS~A052775F$"$K(-F>6, F^hl7%FB<$F]i`m/F_s-FL6(FNFBFBFBFBFBFis77F+F+F2F\`n"$@%"%">%"&AV%"'iw["(Ax_&")( )=4k"*Q,fc("+RvRi!*"-&y7h')4"".0gxGbM""/?SE\?i;"0QS?\H)o?"1+[(\w!y"f#"2@`Ki(\hl K"3K#**fCGPc8%"4AsA$)eZ28E&"5D1VK*pt_2s'F^rF^rF^rF^rFavQ,EIS~A052781F$"$Q(-F>6, F^hl7%FB<$/FB-FL6(FNF_sF_sF_sF_sF_sF`jxFis77FinF+FhpF[amFa_mF^a[n"&wu$"'!\5%"(I `j%")X!*e`"*S^6J'"+cp([a(",v&[6K"*".NJ?zo6""/I1k97y8"0gB@$H]8<"1r%)e@)>Z9#"2g@w nf#>+F"3v)fRq)p<6,F^hl7%FB<$F\[enF\[fnFis77FinF+FhpFddmFbhuF\[il"&E0$"'bcK"(+7e$")]=IS"*Dc#>Y "+0w_t`",IyR"Gj"-g$[H'Hv".]"*R1%Q!*"0+hi)o@$4""1S-bcv/J8"2Ddbh.s+j""3NJW)z+*e1? "45&3R'zSg9[#"5]Yqc#)[%493$F^rF^rF^rF^rFavQ,EIS~A052797F$"$a(-F>6,F^hl7%FB<$Fg` enFaienFis77F+F+FhpF[am"$b$"%vM"&Oe$"'OWQ"(gQC%")&Q0z%"*OVS]&"+m'GbT'",v#)yxc(" -!\c"GP3""0ioF1b@J""1t!>IPy!*f""2,IJ;h**)f>"3&fy[p'\L9C"4!fqy=zJi()H" 5(\U!*pL4)47PF^rF^rF^rF^rFavQ,EIS~A052798F$"$b(-F>6,F^hl7%FB<%F^cxFca_lFciclFgn 77F+F+FhpFjhrFg_`l"&9C""'_NH"(1[P)"*+Y1!G",KyFZ2""-#f()yfl%"/#zA)[*zC#"1+GK]/1( >""2g*)3P%f"*pp"4K_&=qUrR0W"6W"oA44'[$3/I"8!G(p`l1L&4Q)>#":g\1H9cbQb4&=<"gT3nR:D+n?$[.g7"@C=)*z.nvMz=v/jQ<"F^rF^rF^rF^rFavQ,EIS~A052802F$"$g (-F>6,F^hl7%FB<%F_eglFca_lF^]wFis77F+F+F5FdpF:Ff`mF\`nF[eqF[_m"$(f"%D9"%=M"%N#) "&5*>"&(G["'7u6"']hG"'r()p"(.%4<"(e#)=%")@iF5F^rF^rF^rF^rFavQ,EIS~A052806F$Fcca l-F>6,F^hl7%FB<%FcdwF\[`lFagbmFis77FinF+F5FhpFddoF[`nFc^m"$K$F]afm"%gL"&>5""&An $"'vQ7"'\CU"(`NX""(;3/&")o%*f<")vU"='"*%e_#=#"*\lAu("+FP/eFF^rF^rF^rF^rFavQ,EIS ~A036249F$Fgg`l-F>6,F^hl7%FB<%F^adnF][en/F_s-FIFbjxFis77FinF+F+F4FiqF`crFjdmFgh el"$(*)"%3K"&e;""&VK%"'xC;"'>#='"(*puB"(T0?*")6,F^hl7%FB<%F^]w/FBF`egl/FbsF`cfnFis77 FinF+F+F5F2FcuFcirF\[qFecr"%B<"%Uf"&<3#"&QR("'6cE"'DK'*"(n;_$")'pmH"")iy.["*Sj$ *y""*(>[(p'"+;$)o6,F^hl7%FB<%FffalFifal/ FbsFcddmFis77FinF+F5FhpFcuFb[sFgfclFad[l"%RQ"&_n""&,Z("'F$Q$"("=`:"(">3s")*QoP$ "*bOYf""**)>He("+p3*yi$",%es0X<",3p3UV)"-e%)Q%R4%F^rF^rF^rF^rFavQ,EIS~A052815F$ F_iv-F>6,F^hl7%FB<%FadwF_eglF^]wFis77FinF+F5F4F2F]blFg`mF^_lFjjpF[`lFi\`l"%!4"" %ND"%qe"&uP""&$QK"&2m("'4<="'lGV"(dO.""(peZ#F^rF^rF^rF^rFavQ,EIS~A052817F$"$#y- F>6,F^hl7%FB<%Fjg^nF][enF_cfnFis77FinF+F+F4F]blF[`nFe^m"$$f"%:D"&v4""&Z)["'r5A" (')Q,""(IDq%")Ft,A"*0b#R5"*R,+%\"+vjmiB",*RU9O6",f`W)*[&"-R)4rTm#F^rF^rF^rF^rFa vQ,EIS~A052818F$"$$y-F>6,F^hl7%FBF]hfnFgn77FinF+F5FcdmFcco"%+L"&%[u"(k[0#")[%)* p'"++9e?D"-?$yFZ2"".7Nww:7&"0/N(eQf(p#"2+k>aeyhb""3SaC>@B)yv*"5![Gy_Sr&43m"7/.H wauxN`1["9gP>J@1iZst$";!GpJs0,Sp>6,F^hl7%FB<%F_eglF^[`lFj\`mFis77F+F+F+F4F2Fcdm F_blFb^mFf\`lFeg^m"%#\""%,R"&y-""&8t#"&UI("'&e'>"'Z=`"("*fW""(#G[R")CN#3"")HhxH F^rF^rF^rF^rFavQ,EIS~A052827F$"$#z-F>6,F^hl7%FB<%FcdwF\[`l/FbsF[jcnFis77FinF+F5 FdpF]blF`crFablFd`el"$$e"%i<"%Ta"&Uq""&sS&"'ML<"'f1c"(1t#="(qb*f")Nry>")EKkl"*K vx=#"*26=K(F^rF^rF^rF^rFavQ,EIS~A052829F$"$%z-F>6,F^hl7%FB<%FffalFifalF[\gnFis7 7FinF+F5FdpFgimFhjpFa_lFcc\m"$b*"%NK"&S7""&f&R"'<79"'f!4&"([A&="(s.z'")OD1D")'G VI*"*8IAZ$"+wux,8"+Fd$3!\F^rF^rF^rF^rFavQ,EIS~A052831F$"$'z-F>6,F^hl7%FB<%F^aem /FBF^__lF[\gnFis77FinF+F5FhpFcuF\\lFd`el"$Y("%;H"&];""&=u%"'%z&>"'Z$=)"(t^X$")- br9")QM9j"*r=ss#"+Nxp%=""+%[5E<&",[Mi(oA",ZB==***F^rF^rF^rF^rFavQ,EIS~A052836F$ "$.)-F>6,F^hl7%FB<%FadwFbddmFbhxFgn77FinF+F4FcdoFia\m"%/^"'u`6"(sf=$"*)oYR5"+;q _7R"-gti%)o;".s;^f^&z"0g0O#)p5>%"2%)>puC/#=C"4!)o!f'=!Rc;:"6gL1!y\9.RF^rF ^rF^rF^rFavQ,EIS~A052842F$"$4)-F>6,F^hl7%FB<%F^hxFbhxF^]wFis77F+F+F5F4F2F]blFf` mF`rF`_lFf^m"$[$"$^("%h;"%&p$"%N$)"&>*="&5L%"&k'**"'+2B"'Fj`"()H_7F^rF^rF^rF^rF avQ,EIS~A052843F$"$6)-F>6,F^hl7%FB<%FadwFbhxF`jxFis77FinF+F5FhpFb\lFc\lFf]rFj`e n"%V;"%>f"&t<#"&z7)"'$[2$"(_`<""(hT`%")**pi<").F**o"*\7kr#"+kV9v5"+n[FvU",vA#=2 6,F^hl7%FB<%F^hxF^^jmFcdwFis77F+F+F4FiqF celFc_sFe_ilF`a`l"%=BF\\s"&.d#"&`r)"'u&)H"(/6.""(jse$")_'eD"")2[@W"*4$Qk:"*lR(f b"+yr"Q)>"+qRJ/rF^rF^rF^rF^rFavQ,EIS~A052855F$"$B)-F>6,F^hl7%FB<%F\[enFc]hlFbhx Fis77FinF+F5FdpFgimFhjpFfinF`eal"$w)Fc_fl"%*p*"&1J$"'vX6"'Z0S"(LLT""(E]-&")#*y) z"")B"pZ'"*P$QWB"*3">D&)"+%Q"48JF^rF^rF^rF^rFavQ,EIS~A052869F$Fdgm-F>6,F^hl7%FB <%Fg`enFj[nFdhlFis77F+F+F5F2FcdmF]hrFgaoFfjgmFeer"%zO"&,;""&Iq$"'i#>""'DtQ"(ZmE ""(k#oT")l:z8")"4ce%"*1V8`""*vNS8&"+U]RF6 ,F^hl7%FB<%Fc_^nF`a_lF^hxFis77F+F+F4F:F[tFhgqFgaan"%]=F^w"&nC$"'J'R""'!G3'"(")) zE")#\;>"")2uT`"*X#=6C"+"GI]4""+Ss%)**\",YodQH#"-/6z!p0""-\7/g))[F^rF^rF^rF^rFa vQ,EIS~A052872F$"$V)-F>6,F^hl7%FB<%Fg`en/F_sFg_alF[ednFis77F+F+F5Fe]lFg`mFd]q"$ .$"%y6"%vY"&^*="&Dz("'UYK"(=nO""(vl!e")z>'[#"*%**yr5"*bf![Y"+*Q=k-#"+IpEw))",>% 4]/R"-6Ov1C6,F^hl7%FB<%F^aemFd^gnFagbmFis 77FinF+F5F2Ff`mF]`nF]jjl"%16"%>X"&3*="&I0)"'W"[$"(#\B:"(jht'")&RY+$"*e9/N""*4Z& 4h"+._=!y#",1&fmr7",'3ITVe"-?U()='p#F^rF^rF^rF^rFavQ,EIS~A052884F$"$d)-F>6,F^hl 7%FBFb]hlFis77FinF+F5FhpFb\lFc\lFd\rFd]v"%=:"%=`"&*)*="&c)o"',HD"'Z)Q*"(#3"*:I'>u"+[GblG",@kp86"F^rF^rF^rF^rFavQ,EIS~A035052F$Faadn-F>6, F^hl7%FB<%Fg`enFjddnF[ednFis77F+F+F5F2FcuFcirFf\v"$$\"%B;Ffjr"&D'="&tV'"'V[A"'Y Dz"(Va"G"):,25")SMBO"*8P1J""*g.Kw%"+#[,%Q<"+>+vojF^rF^rF^rF^rFavQ,EIS~A052890F$ "$m)-F>6,F^hl7%FB<%FbddmFc]hlFbhxFis77FinF+F5FhpFcuFb[sFjczFjdgn"%'o$"&.f""&.,( "'USJ"(wgU""(g![l")&p_.$"*'3P=9"*f"pum"+<-PgJ",vVCX]"",qNRq>("-P:WmdMF^rF^rF^rF ^rFavQ,EIS~A052891F$"$n)-F>6,F^hl7%FBF_a_lFis77F+F+F4F]blF[`nFe^m"$*e"%$[#"&Y2" "&?u%"'oE@"'Cj'*"(S&RW")'G(e?")%[Pi*"*'H7IX"+;(ya9#",8?#f@5",p$Qz()["-tM,i[B".G !HB!H8"F^rF^rF^rF^rFavQ,EIS~A052893F$"$p)*&-Fc[l6#Ff\hlF+F*F+7#-F>6,F^hl7%FB<%/ FBF_aem/F_s-FF6$FBFjhlF^adnFgn77FinF+F5FjqFcgq"%b:"&#oE"':#e&")[9y8"*!Q4AR",50& =k7"-D]s)>b%"/k^Xt$4"="0\&zO$[&))y"2ABZ*p!>Vt$"4SU+)))\;()3>"6+cf%R0yT*y/""7>wh 4+8!Hi&[h"9q#\&yB3s$e%4SQ";R24KAM\RGU\VD"=?_m\7O7qCB&R3y"Fi_hnF^rF^rF^rFavQ,EIS ~A035051F$Facx,$*$FjcflF;F;7#-F>6,F`el7%FB<#/FB-Fe^l6#-FV6$FNF`bzFgn77F+F+F5F2F celFhyFdelFeelFfelFjh_lF`i\nF[ean"++S-&=("-+gTSX7F_j_m"/+?^G4YW"0++'\aD:%*"2++w &oX7M@"3++Cek*)*=7&"5+?>OLk!Q%*G""6++gHZ9"G11MFgahn<(FdoF^uFdflF]glFdzF^efl*&-F ^cl6$,$*&,,F_e_lF+Fa]oF\tFdclFhddlFfclFbbbnFcclFbeclF+FhclF+#F+F]fflF^fflF+F_gl F+,$*&FifflF;F[gflF+F;FavQ,EIS~A052632F$Feg^m,&F`[`mF+Ff[fmF+7#-F>6,Fav7%FB<%/F bs-FVF`hvFf\alF\[`lFgn77F+F4F2FgimFf`mF[`nFidmF\\oFfi[mF^\o"%M5"%f?"%3T"%0#)"&) R;"&$yK"&_b'"'*3J""'i@E"'2V_"('f[5Fgchn<&FdoFdfnFd\x,*FhgoF+*&,&F]qF4F4F+F+FfuF +F+F\]lF+F[rF+F^rF\\`mFavQ,EIS~A006127F$Fjhv*&FieoF+,(F;F+F*F5FjxF5F;7#-F>6,Ffa n7%FB<#/FB-Fe^l6#-FL6%FdfoFc`mFNFis77F+FinF5FdpFjqF[tFfinFhy"$c'"%#z""%'*["&wL" "&Wl$"&S)**"'oFF"';_u"(of.#"(oBc&")sm>:")!3=:%"*/&HM6F`ehn<%Fdo,(FeqF5FfuF5FjzF ;F[p-F^cl6$,$F\h`mF_em/Fccl-F-6#,(F;F+F0F5FjemF5*(,&FboF+Fhd^nF+F+,(F+F+Fhd^nF; *$,&Fhd^nFfrF\vF+F5F[yF;)FgghnFfjlF+FavQ,EIS~A028860F$Fec_l*2F_hpF;F]_oF;FahpF; ,&F;F+*$F*Fb\lF+F;Fba`lF;F]`\lF;FchpF;F_ejmF;7#-F>6,Fhcm7%FB<#/FB-FL6*FbipFg_oF hip-Fe^l6#-FL60FNFNFNFNFNFNFNFNFNFNFNFNFNFNF\b`lF^a\lF[jpFiejmFis77F+FinF+FinF+ FinF+FinF5FinF5FinF4FinFdpFinFhpFinF2FinFiqFjghn/Fjcq"%A=Fbf\l/F^eq"%i@/F`^qF\_^n/Fg]qFbgl/F^]qF\gn/Fi^qF_d[l/Ff_sFihv/F[^sFe e_l/Fc]sFhe[l/Fd^sFc_hm/F`[sFj]u/F_jrF]fq/F]irFf^q/FfirFjjpFcf\lFff\lFhf\lF]g\l Fjcel,^[l!0+g]NjN]%F+F]q!/[bIe*Ha#F``s!$o)FeqFeenFfuFignFjz"'+k(Fegs",SMvZ'oFigs",?rj w;'F]hs",gD)G@lFahs"+g*yI6"Fehs"+++1m=Fihs"+!oz9X"F]is"*g0$\XFais"*?2^S)Feis"*g ")4C'Fiis",?vA*e9F]js",SA[h3"Fajs",?>EME"Fejs"++cU!e'Fijs"+!['zTyF][t"++_$GZ&Fa [t"++sexOFe[t"+?&\)3XFi[t"+?$=x'HF]\t"+g8rnBFa\t",gx86r#Fe\t",!3I7DCFi\t",g@Gj: #F]]t",?26Hn"Fa]t",+cUa!>Fe]t"+?f:m#*Fi]t",!o6J8IF]^tF;F^^t"*+3Xa"Fb^t"*S1)oAFf ^t"*!)3<-"Fj^t")?"4a'F]cp")++KSFjbp")?(yP#Fe`o")+cI8Fa_t"*+)G\KFe_tFd_inFi_tFe_ inF]`tFc_inFa`tF`_inFe`tFa_inFi`tFb_inF]atFh^inF`atFe^inFcatFd^inFfatFa^inFiatF c^inF\btFg^inF_btFf^inFbbtF^ainFebtF\ainF[ctFi`inF^ctFj`inFactFh`inFcdtF\^inFfd tFi]inFidtFfhblF\etF[^inF_etF]^inFbetF]`inFeetF\`inFhetFeenF[ftFignF^ftF[`inFaf tFaainFdftF_ainFgftF`ainFjftFfainF]gtFdainF`gtFf`inFcgtFeainFfgtFcainFigtFbainF \htFh_inF_htFj_inFbhtFi_inFehtFg`inFhhtF^`inF[itFb`inF^itF]ainFaitF_`inFditF``i nFgitFe`inFjitF[ainF]jtFd`inF`jtFc`inFcjtFa`inFfjtFf_inFijtFb^inF\[uFg_in,T-F^c l6$,$*2)Fccl,&F7F+F]qF;F+FhuF+F\[lF+F]rF+F\clF+Fg\oF+Fd`oF+Fg`oF+#F+".+S=<:^$F] cwF+-F^cl6$,$*,FcclF+,&FfclFj[oFdinF+F+F[huF+FhuF+F\[lF+#F;"(?&*4#FabuF+-F^cl6$ ,$Fd[[n#!%r?"'OZ))Ff[\lF+-F^cl6$,$Fijjm#F+"(!3)R)FabuF+-F^cl6$,$F_[[n#"*">K$p'" -+glA3DF]cwF+-F^cl6$,$Fijjm#F+"(!oBWFf[\lF+-F^cl6$,$*,FccuF+FhuF+F\[lF+F]rF+F\c lF+#"&F(=",Si!H.5F]cwF+-F^cl6$,$*0Fh^_mF+FhuF+F\[lF+F]rF+F\clF+Fg\oF+Fd`oF+#F+" ++_vg$)F]cwF+-F^cl6$,$*(FcclF+,*"%o7F+Ffcl"%l=Feao"%T:Fa]o!$L%F+FhclF+#F+"&WH)/ Fccl-F-6#,,F[_\lF+F]boF;F_dlF5FjemF;F+F+F+-F^cl6$,$Fijjm#"*,**)H9".+g$f$\]"F]cw F+-F^cl6$,$*(FcclF+,&F+F+Fa]oF+F+FhclF+#F+F]_m/Fccl-F-6#,(F[_\lF+F_dlF;F+F+F+-F ^cl6$,$Fg\[nFj][nF[^[nF+-F^cl6$,$*(,*Fa]oF4FfclF4F5F+FeaoF5F+F]buF+FhuF+#F+Fej] nF^i`lF+-F^cl6$,$F_cn#"+PT\aX",+?(y)Q#F]cwF+-F^cl6$,$*(FcclF+,.F+F+FfclF4FcfuF+ FafuF+FjduF+FeaoFboF+FhclF+F_][nF`][nF+-F^cl6$,$F_cn#"%ZOFhb]nFf[\lF+-F^cl6$,$F d[[n#"-4#)=EER"/+?b^X`5F]cwF+-F^cl6$,$*(FcclF+,*F\_blF+FfclFc_\lFeaoFgdmFa]oFhz F+FhclF+#F;"%+vF^i`lF+-F^cl6$,$*(FcclF+,.F2F+FfclFhpFa]oFdpFjduF5F\euF+FeaoF4F+ FhclF+#F+"$'o/Fccl-F-6#,0FeeuF+Fe^^lF+F[_\lF+F]boF+F_dlF+FjemF+F+F+F+-F^cl6$,$F _[[n#Fc_\lF`]amFf[\lF+-F^cl6$,$*(,*FdqF+FfclFabnFa]oFdqFeaoFiqF+F]buF+FhuF+#F+" &[w#FeeinF+-F^cl6$,$*.Fh^_mF+FhuF+F\[lF+F]rF+F\clF+Fg\oF+#F]fxF_`uF]cwF+-F^cl6$ ,$*(,&"%nDF+Ffcl"%n9F+F]buF+FhuF+#F;"(;'z;FabuF+-F^cl6$,$*(FcclF+,&Ffcl"&6Z%"&! eIF+F+FhclF+#F+Fb[jnFabuF+-F^cl6$,$*(FcclF+,.F5F+FfclF5Fa]oF5F\euF+FjduF+FeaoF+ F+FhclF+#F+FghzFhj`lF+*&,&F^binF+Fb[lF^binF+F]qFe]lQconumber~of~ways~to~make~n~ cents~with~coins~of~2~8~12~14~18~20~24~30~centsF$Q,EIS~A008582F$F[_m*&-Fc[l6#-F -6#,(F[]hlF+*(F\]hlF+F*F+F0F+F+F0F;F+F*F+7#-F>6,F^hl7%FB<%Fa`hnF^adn/F_s-Fe^lFd `hnFgn77FinF+F5FcdmF[jn"%XJ"&w*o"(**p%=")SMTe"+@2uI@",+3Uh!))"."*4Yio1%"0[Weogb 2#"20D6mUO+;""3kS5%*y_kYq"5vB)[j"*p3Ei%"7'p$4ScCm?)oD$"9Ti)GU^qYB$z_C";?Ng1P]>[ KrKm>"=6lwzfZ0CTCe#=n""?++%)zh%e%eU"3J6,F^hl7%FB<%F^hxFcdwFddwFis77F+FinF +F5FdpF2FjqFd]mFbbnFdinFfinFf]rF\ijlFhbr"$o&"$!))"%y8"%5@"%gK"%U\"%FvFe^jnF^rF^ rF^rFavQ,EIS~A052847F$"$:),$*$FafmFdpF\e^l7#-F>6,F`im7%FB<#/FB-FF6$F_gm/FSFdpFg n77FinFinFinFinF+F]blFf^qFgd_l"%pnFadgm"'!oB("(+&4%)"*w!e_5"+))[,99",'4`PJ?"-+/ E*)4J".CQq&*p0&"//f()[x2()"1)[Fd(RJ$e""2C%>x+a7KI"3w<\Sd2;,hFi_jn<(F_uF[pF^pFfa vFhbl,,*&,,FcqF+F\qF2F`qFdpF+F+F]qFdpF+FfuF+F+*&,*F]q!#GF\qFe_mF`qF[yFhhdlF+F+F jzF+F+*&,(F`crF+F]qFcelF\qF2F+F[rF+F+*&,&FgdlF+F]qF[yF+FjblF+F+Fe\oF+F^r,$F`cvF _emQ\oPermutation~with~exactly~4~cycles,~Stirling~numbers~of~first~kindF$Q,EIS~ A000454F$Fbbn-Fc[l6#Fcg_l7#-F>6,Fav7%FB<%F^hxF_eglF`hxFgn77F+F+F4F\_lFfd[m"%@F" &r"\"($[%3"")HdCG"*`jX[)",J`a*"=@FY_"yy =7m,+\1(F_bjn<%FdoF^u,(FeqF;Fg`[lF+FjzF+F^r,$**F[vF\vF`ffmF+F]qF]vF^]mF+F`[lFav Q,EIS~A001517F$Fcbl,$*(FdcmF;Fj`lF;FeazF;F;7#-F>6,Fhcm7%FB<#/FB-FL6%F^dmFhalF^b zFis77F+FinFinF+F+F+F+F+F5F5F5F5F4F4F4FdpFdpFdpFhpFhpF2F\djn<,FdoF[pF^pFi\lFhbl FcbzFicelFfcclFjcel,6,F^hl7%FB<%F^gmFc`yFiefnFgn77 FinF+F4Ff`mF^jn"%CM"&3!p"(ciq"")Si&*\"+wt\*o"",_ZD*zk".Kox1*yF"0#>?\hc<8"1Ca"=B 0M%o"3sIG*=1YT'Q"5[+i%*Ra""9!)[NB]\9xjP"3"":Saspj#*)>@e#G1)"g<-)y$G%[na/U0J`FiejnF^rF^r,$*,F[vF\vFdpFfr*(,&F+F+*&,&-Fc[l6#F `[lF+F;F+F+FhgjnF;F;F+,&FhgjnF;F+F+F+,&F;F+FfgjnF+F;FfrF]qF]v-Fc[l6#,$*&,&-F_jl 6#FggjnFdpF;F+F+F]qF+Fg[lF+F`[lFavQ,EIS~A052851F$Fgdu,$*(F*F+Ff]lF;F\]_lF+F;7#- F>6,F^hl7%FB<%F\[`lFj[nF^_hlFis77FinFinF+F4F2FgimFd]mF]_lF`blFf^qFe^mF\`vF_]v"$ *y"%?9"%,EF`[en"%2*)"&=m""&>7$"&9)eFdhjnF^rF^rF^rFavQ,EIS~A052825F$"$!z*&F`hpF+ ,(F;F+Fj]pF+F*F+F;7#-F>6,F`al7%FB<#/FB-Fe^l6#-FL6$FNFeipFis77F+F+F+F+F+F+F+F5F4 FdpFhpF2Fe]lF:FjqFcuFc\nF\`oFj`mF^_lF`]lFeijn<*FdoF^uFgblFi\lFhblFcbz,(FeqF+Fe\ oF+Fb`oF;Fjcel-F^cl6$,$*&,.Fccl!$D'Fdao"%+XFa]o"%+aFdclFdenFfcl"%wxF]dcnF+F+Fhc lF+#F+"&"y\/Fccl-F-6#,(F;F+F]boF+F0F+*(,&F;F+*$-F-6$Fb[[o$"+()f*3y(FgdlF2F+F+,& Fe[[oF1Ff[[oF;F;),&F+F+*$Ff[[oFhpF+F]qF+FavQ,EIS~A005708F$"$z$*,Ff]lFboF_hpF;Fj `lF;F[`\lF;F]`\lF;7#-F>6,Fhcm7%FB<#/FB-FL6(Fd^lFd^lFbipFhalF[a\lF^a\lFis77F+F5F dpF2F]blFb\lFf`mF^blFddmF]hrFgdmFidmFfinFfdqFb[rFd`nF]d^lFi^mFjgalFi]_lFa^vF`\[ oV"'+SQFcq#F\_l"&+%QF\ q#"%f()F[`[oF``u#"%2H"&+c#F`q#Fd_lF^wFd`u#F\_l"&+G"F]au#F+F[`[oFaau#"%.>"&+o(-F ^cl6$,$*(,4FerF+FcclFb_\lFdclFb_\lFfcl!#KFdduFe]lFa]oF]a[oFdaoFb_\lFeaoFerFbduF e]lF+F]buF+FhuF+#F+"&+g"/Fccl-F-6#,.Fe^^lF+F[_\lF5F]boF5F_dlF5FjemF5F+F+F+-F^cl 6$,$*&,6FcclFi^jlFdcl"$$*)Fddu"%#"'<56"'+!G"F+,$F[asFjb[ oQ[onumber~of~ways~to~make~n~cents~with~coins~of~1~1~2~4~10~20~centsF$Q,EIS~A00 1307F$Fie[m,$*$,(F;F+F*FdpFjxF+F;F;7$-F>6,Fav7%FB<#/FB-Fe^l6#-FV6'FNFNFNFNFdipF is77F+FdpFabnFa^m"$0$"%#H""%ta"&%=B"&4#)*"'?gT"(*Gi<"(w^Y(")$*HiJ"*[r&R8"*&e^uc "+)[jPS#",Pb]#=5",OcyLJ%"-"3[wr#="-gzP/Sx".@*f^tyKFbc[o<%,(FeqF+FfuFdpFjzF;FdoF hbw-F^cl6$,$F[cwF[^fl/Fccl-F-6#,(F;F+F0FdpFjemF+,$*&,(FiqF+F_^nF[y*$,&FboF+F_^n F+F5FboF;)F^f[oFfjlF+F;FavQ,EIS~A001076F$"$)R-F>6,F`el7%FBFic[oFgn77F+FdpFj`mFh ]v"%?t"'S]:"(g0%R"*gt%o6"+!)oyfR"-+wLl4:".+KK%*\R'"0+oNPf)zH"2+))yVUYZ^""3+%yY! p%R:M)"5+?bJ!>&f%p%\"7+Sev=****)RL9$"9+g&)4w/%)QUYI@";+g0Hw-8pFX@M:"=+!oz!zbcik lm#)p6">++sU,&Qv>QFSQ:%*"@++WXk,+y:^[NTo(zFbc[o<%FdoFhbw,(F\^ilF+F^_imF+FjzF+*& Fce[oF+F_glF+Fje[oFavQ,EIS~A052630F$"$w&*&FafmF4F*F+7#-F>6,F^hl7%FB<$F^gmF`[gmF gn77FinFinFinFinFcelFg^mFg\cm"%]%*"&_z("'G"4("(Su3(")+#)Rx"*suD=*",G@gr<""-oB+^ A;".S]`/PR#"//6f_qkP"0S9q'R8*G'"2g*3CWiJ76"3g*3soyEm2#"4+kse7&*p93%F\h[o<(F_uF[ pF^pFfavF]gl,**&,0FcqFdpF``sF;F]qFa`lF\qFdqF[asF^vF2F+F`qF]blF+FeqF+F+*&,,F[asF 4FcqFjqF`qFdpF\qFheclF]qF2F+FfuF+F+*&,(FcqF^vF`qFhzF\qFdqF+FjzF+F+*&FgayF+F[rF+ F+F^r,$*&F^[nF5F]qF;F4FavQ,EIS~A052757F$"$9(**FdcmF;FeazF;Fb]glF;FgbelF;7#-F>6, Fhcm7%FB<#/FB-FL6&F^dmF^bzF[^glFccelFis77F+FinFinF+FinF+F+F+F+F5F5F+F4F5F4FdpF4 FdpFhpFhpFhpF^j[o<7FdoF[pF^pFapFdcclFi\lFcbzFicelFecclFbf\lFjcelF\del/Fa`rF+F[d cl/F`arF4/FfarFhp/Fg`rF5/FiarFhp,T!&E1"F+F]q!%^9FeqF2FfuFcelFjzFihpFe\oF\_mF[rF [[qFjblFhdwFb`oFa_mF\qFgfyF`qF;FahsFihpFehsF2FihsFcelF]isF\_mFaisF[[qFeisFhdwF^ ^t"$;&Fb^tFd]vFf^t"$k&Fj^tF\gnF]cpFb[\oFjbpFa[\oFe`oFd]vFa_tFa_mFddclFhdcl,4F`q #F+FaafmF\q#F5FdbrF]q#"$"RFaafm#F[cjlFejdlF+-F^cl6$,$*&,,FcclF;FdaoF4FdclF+F5F+ FfclF5F+FhclF+#F;FcirF[dglF+-F^cl6$,$*(FcclF+Fj`blF+FhclF+F]czF^czF+-F^cl6$,$Fe g`lFfaglFfemF+-F^cl6$,$*&,&F+F+FcclFabnF+FhclF+#F5F_[qFfemF+-F^cl6$,$*(FcclF5Fj `blF+FhclF+#F;F\`oFgaglF+,$F`qFd[\oQennumber~of~ways~to~make~n~cents~with~coins ~of~3~5~7~9~centsF$Q,EIS~A008678F$Fhe[l,$*(F*F+Ff]lF;Fa\wF+F;7#-F>6,F^hl7%FB<%F \[`lFj[nF^]wFis77FinF+F5F4FhpFe]lF]blFb\lF\_lF`crFbbnF^_lF`]lFidmFfinFheqFa]rF\ j\lF^fmFbhelF]d^mFd]\oF^rF^rF^rFavQ,EIS~A036469F$"$u(,$*$,(FboF+FfglF+F*F+F;F;7 #-F>6,Fav7%FB<%F]dhnFca_lF_eyFgn77F+F5F:F[\oFj]_l"%@i"&&G%)"(bAL"")"pmS#"*(H+"* [",L'oU/6"-613FVF".V4)\ULu"0$4mBM3#=#"1"=k#>>C)*o"3.h%p7w:lL#"4>5]Xdfd;W)"6D](G HlLt_SK"8DAy%=37;d7<8"92$R)=#HMz=H4l&";&4Lvd6'[%pL[?b#F^^\oF^rF^r,$*&,(*&-Fc[l6 #,&-Feix6#FcgyF;F5F+F+,&F^`\oF+FboF+F+F+F^`\oF+FboF+F;)F]`\oFfjlF+F;FavQ,EIS~A0 06155F$Fhg\l,$*&FhfdlF+Ff]lFboF;7#-F>6,F`el7%FB<#/FB-FL6$Fd^lFagdlFgn77F5FhpFb\ lF\\lFfi[m"%g:Ff]el"&!o&)"'!3m("(![?w")+CY$)"*++#z**",+K/LH""-+Kg$e!=".+s-FDq#" /+S9aK:V"0++3YwHK("2+?bR[VgJ""3+?RBXd#p\#"4+?6in6\u)\"6++_&f^jy9Y5Fc`\o<&FibnF^ fhlF[gx,&*&,(F]qFa`lF\cuF+F\qFboF+FeqF+F+*&,&F4F+F]qF5F+FfuF+F+*&Fbb\oF+F_glF+F _amFavQ,EIS~A052649F$"$'f,&*&F^cbmF+FfglF+F+F^cbmF;7#-F>6,F^hl7%FB<%F^cxFc`yF_e yFgn77FinFinF5F:F[amFa`\m"%Y9"&$*=""'We6"(*4-8")!Hol""*PzBN#"+3#)*\o$",V&)R8J'" .%\IK,t6"/T"oeT4N#"0KZw$[e`]"2ZJoHUB(f6"3ew">lx+(HG"40,)QAQ4y9t"6;IR%R.\-*o*>Ff b\oF^rF^rFhdbmFavQ,EIS~A052846F$"$9),$*&Ff]lF5,*F;F+F*F4FjxFboF9F+F;F;7#-F>6,Ff an7%FB<#/FB-Fe^l6#-FV6$FN-FL6&Fd^lFd^lFNFNFis77F+F+F5FhpFjqFh`mFcbnFf_lF]gilF]e u"%(*="%5W"&_-""&LQ#"&0a&"',)G""'E%*H"'"3'p"(#>=;"(S=w$"(<_u)F_d\o<&FdoF^uFdfl, *FeqF+FfuFboFjzF4F[rF;-F^cl6$,$*&,(F;F+FcclF7FfclF5F+FhclF+Fj``m/Fccl-F-6#,*F;F +F0F4FjemFboFj]oF+,$*(,&F;F+-F-6$Fdf\o$"+!4(f,VFgdlF+F5,(*$Fhf\oF5FdpFhf\oF^v*$ Fhf\oF4F^vF;),(F4F+Fhf\oFboF]g\oF+F]qF+F;FavQ,EIS~A034943F$"$/*,$*(F*F+,&FhhdlF +F*Fe]lF+Ff]lFboF;7#-F>6,Fj]l7%FB<#/FB-FV6$F^dvFg^lFis77FinFcdmFcdoF]_lF]`oFjhr F`]lFhdmFc_sF`_lFbcoFbblFb_lF]fqFahqFd_lF\\oF^_rFf_lFcgzFdj\lFcg\o<&F_u,(FeqF+F a`lF+F]qF7/F_pFcdo/F\pFcdm,&Fe]lF+F]qFiqFgdvQ)~8~n~+~7F$Q,EIS~A017149F$"$i*,$Fc ]mFiq7#-F>6,Q/Multiples~of~8F$7%FB<#/FBF^dvFis77FinFiqFcuFcelF[tF[amFeclFb[sFgc lFa^mFijpFfinFhaqF\dqFginFcenFgaoFb`nFe^mF^]]lFjinFgh\o<%F_uFj^z,&FeqF+F]qF7Fgd vFgdvF[i\oQ,EIS~A008590F$F\t,*FfrF+F\\[lF\vFjxF;F*F;7$-F>6,F[s7%FB<%Ffh_l/F_s-F V6%FBFbsFNFa_[lFis77FinFinF+FdpFjqF[amFe^mF[][lF\][lF]][lF^][lF_][lF`][lFa][lFb ][lFc][lFd][lFe][lFf][lFg][lFh][lFci\o<(F_uF[pFgblFj][lF``o/FbpFjqF^rF`^[lFavFg ^[lFegfm-F>6,Ffv7%FB<%Fg]_lFivFi\[lFgn77FinFinF5FcelF]_mFc_[lFd_[lFe_[lFf_[lFg_ [lFh_[lFi_[lFj_[lF[`[lF\`[lF]`[lF^`[lF_`[lF``[lFa`[lFb`[lFci\o<(Fd`[lFdflF_uF[p Fjgo/FbpF]_mF^rF`^[lFavQ,EIS~A052739F$"$&p*$Ff]lF77#-F>6,F]c_l7%FB<$Fi[nF_ielFi s77F+FiqF_blFcenFh_mFf[gnFfc_l"%KMF_cgm"&S9""&[%>Ffafl"&)Q]"&?v(Ficcm"'W0<"'d^C "'/hM"'+2["'+yl"'I!)))Fhj\o<$Fdo,4FeqFeenF]^tF;F``sFgajnF[as!$A$Fcq!%g>F`q!%pnF \q!&KJ"F]q!&oI"F[gbnF+,$*0FhuF+F\[lF+F]rF+F\clF+Fg\oF+Fd`oF+Fg`oF+#F+Feen,$F]^t Fc\]oQOArrangements~of~n~unlabelled~objects~in~8~rowsF$Q,EIS~A000580F$Febn,$*&F *F5FcboF;F;7#-F>6,QMWords~with~at~least~2~consecutive~b-letters~F$7%FB<&FeeimFg eim/Fjeim-Fe^l6$FheimFa[zF]fimFis77FinFinF+F+F+F5F4FdpF2F:FddoF\_lFh`mFj[oFihpF finFdbnFdfwFi]cl"$1%FefwFg\]o<&F_uF[pFgbl,(FeqF+FjzF+F[rF;-F^cl6$,$*&,(FfclF5Fc clF4FdqF+F+FhclF+F[hwFfdo,$*(F\eoF5F`eoF;FbeoF+F;QLWords~with~at~least~2~consec utive~b-lettersF$Q,EIS~A000930F$Fb\l,$*(,*FboF+F*F+FjxF+F9F+F+Fg_`mF;Ff]lF;F;7# -F>6,Ffan7%FB<#/FB-FV6$-Fe^l6#-FL6$F[^`mFNFd^lFis77F5F+F5F5F5F4F4FdpFhpF2FiqF]b lFddoFabnFg`mFhjpF`rF`blF]^sFbcoFc_lF_^]o<'F^uFdflFibnFac[l,*FeqF;FfuF;F+F+F[rF +,&F+F+Fe``mF+,$**,*FboF+FdfflF+Faa`mF+F]gflF+F+F`a`mF;FcfflF;F[gflF+F;FavQ,EIS ~A052954F$Fgbn-Fc[l6#F\im7#-F>6,Fav7%FB<%F]i`mF^sFb`yFgn77F+FinF5F2Fj`mF^jn"%)o ""&o[""'%oZ""(w8j"")7>#)>"*!3tDE"+7nFkP",wTgW!e"-sYg`w&*"/?"R%R(Go""0)GqczUPJ"1 [gqQ*=R='"3;Sf6,Q-Powers~of~13F$7%FB<$Ff\al/FB-FL6/F_sF_sF_sF_sF_sF_sF_s F_sF_sF_sF_sF_sF_sFgn77F+FddoF\j\l"%(>#"&h&G"'$Hr$"(4o#[")<&[F'"*@2t:)",t$*\/1" "-\=\ey8Fb\im"/"[A^3)HB"0`Af1^(GI"1*G*p&Qwt$R"2d249I*e=^"3T)zJ=4mTl'"4LzL"Q>fT] ')"6HJRd>&pSbC6"7x16YaP!H?>Y""8,)Q%*z!)[xj\+>F^a]o<$Fdo,&FfuF+FeqFheclF^r,$*,F[ vF\vF5FfrF]qF\v-Fc[l6#*&,&F+F+-F_jlFh`rF+F+F]qF+F+FigyF+FfrQ]oArrangement~of~n~ distinguishable~objects~in~13distinguishable~urnsF$Q,EIS~A001022F$Fi]cl*&FieoF+ FhiblF;7#-F>6,Fav7%FB<#/FB-Fe^l6#-FL6$FdfoF[^`mFis77F+F+FdpFddoF^_lFi^rFh]dn"%\ :"%;^"&(*o""&2e&"'=V="'h(3'"(,1,#"(k0k'")$HK>#")VuVs"*AYCR#"*48')Fec]o<%FdoF^uFf[cl-F^cl6$,$*&,&F;F+FcclFiqF+FhclF+F^h^lF\\cl*(,&F[yF+Ffh^lF+ F+Fc\clF;Ff\clF+FavQ,EIS~A003688F$"$>%,$*$,(F;F+Ff[fmF+F*F;F;F;7#-F>6,F^hl7%FB< %FahlF^aemFghlFgn77F+FinF5F4Fh`mF]jq"%Y6"%F))"&+U*"(,q+"")!R9G""*^T6s""+Ocv%e#" ,p+)oVT"-1*4Dq@("/&z&H"3(R8"0;k]2c5m#"1pJ$H,uag&"3)z]+gyV;D""4$)**)=7hA3[H"5+(e ([y_o?7tFie]oF^rF^r,$*(Fb[lF+,(*&-F-6$,(*&F0F+F\]hlF+F+F;F+F0F;$!+&[w*\8FdqF+-F c[l6#Feg]oF+F;*&F[h]oF+Feg]oF5F;Feg]oF+F;),$*$Feg]oF;F;F]qF+F;FavQ,EIS~A052848F $F]euF`io7#-F>6,Q73-balanced~hierarchiesF$7%FB<#/FB-FF6#-FF6$FhhlFjhlFgn77F+F+F 4FjqFihpFdhdl"%rC"&-$>"'%*y;"(Phg"")zPt;"*-%y$)="+&[BkF#",Cv!yOH"-DMCxDS".d\"4> Se"/:u,,]M*)"1&*ya6U!pV""2-)zx[wqAC"3)fd:IPy=F%"43:_1'Q+$*fyF`ioF^rF^rF^rFeh]oQ ,EIS~A000258F$Fidm,$*&Fh\zF;FjfclF+F;7#-F>6,QKWords~with~at~most~1~consecutive~ b-lettersF$7%FB<&FeeimFgeimF]fim/Fjeim-Fe^l6$Fheim1FSF+Fis77F+F5F4FhpFiqFddoFc\ nFj`mF`]lFh_qFe^mF^]blF]elFhg_nFd\nFg[`lFe\n"%"=%Ff\n"&Y4"Fg\nF^j]o<%FdoF]am,(F eqF+FfuF+FjzF;-F^cl6$,$Fd]anFbjyF[\_l,$*(F_bgmF;,&FfrF+F_^nFfrF+FcbemF+F;Q`rWor ds~with~at~most~1~consecutive~b-letters;~Fibonacci~numbers;~integer~F(n)~such~t hat~F(n)=F(n-1)+F(n-2),~F(0)=F(1)=1F$Qbp~Fibonacci;~EIS~A000045;~http://www.ee. surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.htmlF$F:,$*(F*F+Ff]lF+Fc[nF;Fbo7#-F >6,Ffan7%FB<#/FB-FL6$FbghlFc`mFis77FinF5FdpF]blF^blFf^sF[[]lFj`en"%?7"%%>$"%i$) "&#*=#"&9t&"']+:"'OGR"(e%G5"(QDp#"(c"\q")I\X=")McJ["*s>\E"Fg[^o<&F_uFb]nFa[qF]a m-F^cl6$,$F\h`mFddyFi]n,$**Fa^nF+Ffa]nF+Fc^nF;Fe^nF+FboFavQ,EIS~A052995F$"%s5*$ FfglFcu7#-F>6,Q-Powers~of~16F$7%FB<$Ff\al/FB-FL62F_sF_sF_sF_sF_sF_sF_sF_sF_sF_s F_sF_sF_sF_sF_sF_sFgn77F+FcuF_e_lFdjxF`h\nFah\nFdielF`f]n"+'Hn\H%Fhiel".wxi6&*4 ""/;W/'=#fMK9fsjybv":whqu"HYh>e#*37Fg]^o<$Fdo,&FfuF+FeqFerF^r,$*,F[vF\vF5FfrF]qF\v-Fc[ l6#*&,&F+F+-F_jlFaarF+F+F]qF+F+FigyF+FfrQ]oArrangement~of~n~distinguishable~obj ects~in~16distinguishable~urnsF$Q,EIS~A001025F$Fg\`l*$-F-6#,*F0F;F_dlF+FjemF+F* F+F57$-F>6,F[s7%FB<%Ffh_lFc\[lFii\oFis77FinFinF+F5FhpFcuFb[sFchdlFgg`l"%qH"&E<" "&gq%"'79>"'/ty"(+"pK")k[o8")`8pd"*aOrW#"+yWoV5"++%GGZ%",?^/_#>Ff_^o<(F_`oF_uF[ pFgblFac[l,,*&,*F[[_mF+F]q!%?>F\qFcb]nF`qFdb]nF+FeqF+F+*&,*!%'4*F+F]q!&?.$F\q!& K!GF`q!%OzF+FfuF+F+*&,*F]q"&!QT"&33#F+F\q"&K!GF`q"%siF+FjzF+F+*&,*F\q!&?l#F]q!& /2'!&+c%F+F`q!%%y$F+F[rF+F+*&,*"&OT"F+F`qFh^hnF\q"%,`F]q"&9`"F+FjblF+F+F^r,$*,F [vF\v-F-6#,*FjemF_^r-F-6$,(F;F+Fj]oFdpF0F5$"+&)\eaQFgdlF2*$F]c^oF5!#iFdqF+F+F]c ^oF+F]qF]v),(#FginF]_lF+F]c^o#FcuF]_lFbc^o#Fc[[lF]_lF]qF+F;FavQ,EIS~A052708F$"$ j'-F>6,Ffv7%FB<%Ffh_lFc\[l/F_s-FV6%FBFNFbsFgn77FinFinF5FjqFcenFgfoFignF[e_nF`w" ++Ovx5",+)38bU".+!3Y[y="/+#fUl'o"*"1+K#RQeD!\"3++#>w^X*\G"5+?&fO#)eM&*y""7+SYQM ++!o**4AGbRBR$Qo%Ff_^o <(FdflF_uF[p,,*&,,F\qF^hjmF`qFe[alF]q!%'\#F[[_mF+FcqFdb]nF+FeqF+F+Faa^oF+*&,(F\ qF\b^oF]q"&)[:"&//"F+F+FjzF+F+*&,&!%+wF+F]qFbb^oF+F[rF+F+FjblFh^hnFchn/FbpFcenF ^rFhb^oFavQ,EIS~A052738F$F[eam,$*2FfigmF;F_i_nF;FdcmF;F[gclF;F_hpF^vFf]lF^vFifc lF;Fg`anF;F;7#-F>6,F_gcl7%FB<#/FB-FF6$F[\n1FSFe]lFis77F+F+F5F4FhpFe]lFgimFcdmFc \nFh`mF`rFcirFcbnF^`nFadqFcblFadvFi\vF^\\mFjnF\\vF]f^o<8F[banF]banFdoF^uFdflFho F_`oF^[bn/FiarFjn/FfarF^\\m/FcarFi\v/F`arFadv/F]arFcbl/Fj`rFadq/Fg`rF^`n/Fd`rFc bn/F\brF\\v/Fa`rFcir/F^`rF`r/F\bpFh`m/FiapFc\n,fn"*+s78#F+F]q"))=]C&F``sF+FeqFf d_lFfu!%?VFjz!&+W"Fcq"%&**)Fe\o!'!o@"F[r!&!GNFjbl!&g0(Fb`o!'!['=F\q"(/IP&F`q"'l KHF[asFg^nFahsF^h^oFehsFjg^oFihsF[h^oF]isF`h^oFaisF_h^oFeisF]h^oF]\tFfd_lF^^t!' ?BQFb^t!'+wKFf^t!'gDTFj^tFeh^oF]cpFch^oFjbpFdh^oFe`o!'![e#Fa_tFfh^o,BF``u#Fecr" %3Y-F^cl6$,$*&,&F[bvF+FcclF]_lF+FhclF+#F+"$s*FfemF+-F^cl6$,$Fi`blFjhdlFf^bnF+-F ^cl6$,$F_cnFjcglF[dglF+-F^cl6$,$F_cn#F+FgclFf[\lF+-F^cl6$,$*(FcclF+,(F5F+FfclF5 FcclF+F+FhclF+F^fclF^czF+-F^cl6$,$*(FfczF+F]buF+FhuF+#F;FghzFfemF+#")"*yz()"*+' 4C:F+Fd`u#F+F^]amFaau#Fe]lFh[mF``s#F+FjgnF]q#"&V8#"&S=&Fcq#F`_lFd_[lF\q#"%"*RF[ `[oF`q#F\_]l"&gH"F[as#F+Fh^an,$F``sFjj^oQBPartition~of~n~in~at~most~7~partsF$Q, EIS~A008636F$F]\o*&,(FafmF;F\imF+F;F+F+F`hglF;7#-F>6,F^hl7%FB<%F[gmF^gmFjg^nFgn 77F+F5FdpFcuF_aq"$=("%)p'"&7N("'Wf#*")3%\J""*#zBx?"+c='=h$",SMSN&o".k'z4649"/!) 3oGQ?J"0%Q;7-"RS("2c#ofJR&R(="3'*))[fy^eR]"57H=CG*\D]V""6;_'=A1r.H8V"8C9*RVM<_j pk8Fg[_oF^rF^r**F\famF;F]famF;Fd`hlF+Ff`hlF+FavQ,EIS~A052835F$"$-),$*(F*F+,&Fd] lF+F*FiqF+Ff]lFboF;7#-F>6,Fj]l7%FB<#/FB-FV6$F^[[l-FL6$F\`pFd^lFis77FinF\_lFi`mF j[oFdinFhdmFe[qFf^qFhaqF[eqFa`nFdbnF_^rFf_lFf^mF\\]lF[jnF`[`lF[cjlFjczF]bhlFb]_ o<&F_u/F_pFi`m,(FeqF+F7F+F]qFd[wFb`l,&FiqF+F]qFgimFh[[lQ*~11~n~+~8F$Q,EIS~A0174 85F$Fd\n*&FafmF+F*F47#-F>6,F^hl7%FB<$F^gm/FB-FL6&F_sFNFNFNFgn77FinFinFinFinFcel FihpFhyFg\cm"%k!)Fii_m"'+%=&"(+'*)\")+CA`"*!3-Fi"++#*HDz"-+S'G(*3"".+gPX%4;"/+S '[C1a#"0+_r8\#oU"1+?bv(=Gg("3++#R$G)=6V"Ff^_o<(F_uF[pF^pFfavF]gl,&*&,(F\qF;F]qF 5F4F+F+FeqF+F+F\j\nF+F^rFj^`lFavQ,EIS~A052759F$"$;(,$FdcvF17#-F>6,Fj]l7%FB<#/FB -FV6$F`^l-FL6$Ff_pFd^lFis77FinFd]mFi`mFc\lF\\lF]^sFg\qFb^mFccqF[[qFd^mFf]rFihyF f^mFh[]lFhdwFh^mF^[qF`\vFj^mFedglFb`_o<&F_u,(FeqF+F1F+F]qFhzFa^_oFc_m,&F2F+F]qF jqFe`lQ*~12~n~+~6F$Q,EIS~A017593F$"$&**,&-Fc[l6#,$*&F*F5Ff]lF;F;F+F;F+7#-F>6,F^ hl7%FB<%FcdwFc]hlFbhxFgn77FinFinF5F2F_blFhyFgfo"&Sw""'?J="(+o6#")SQ%p#"*++Au$"+ !3M*Gc",S52A6*".Sm4M!z:"/+Kw*ob"H"0+Ce?*38d"2+wp/Q`Q=""3+o^X'y#3'e#"4+%Q?FuLnQf "6+Gx$*G&H*Q+V"Fda_o<'FdflF_uF[pFdz,*Ffg`mF+*&,(F\qF+F]qF+FboF+F+FfuF+F+FabemF+ F[rF+F^r,$**F[vF\v-Fc[l6#F]vF+F]qFgeblFbffmF+FfrFavQ,EIS~A052845F$"$8)-Fc[l6#,( F`[`mF+FfglFboF+F+7#-F>6,F^hl7%FB<%F]i`mF^sFghlFgn77F+FinF5F2F^blFihyFfh\o"%Yo" &mU&"']6,Fj]l7%FB<#/FB-FV6$Fc`imF_camFis77FinFjqFg`mF[tFc\lFdi nF\]sFa^mF^`nFe`qFccqFginF_iqFd\lFi^rF^]]lFf^mFe`nFbg[mF]wF^f[lF`e_o<&F_uFddv/F _pFg`m,(FeqF+FboF+F]qFgdl,&F5F+F]qF]blF[aimQ*~10~n~+~2F$Q,EIS~A017293F$F`i^o,$* &F_hpF+,*F+F+F*F^vFjxF;F9F5F;F;7#-F>6,Ffan7%FB<#/FB-Fe^l6#-FV6%FNFN-FLFa]dlFis7 7F+F4F:Fh`mFbcoF`eal"$W)"%HE"%*=)"&3b#"&b%z"'&\Z#"'C4x"(d8S#"(0+[(")C&*HB")jeds "*.r1E#"*C"yTq"+\(\M>#"+lJRKoFbf_o<&,*FeqF5FfuF;FjzF^vF[rF+FdoFd\xFahz-F^cl6$,$ *&,(FgfyF+FcclFhhdlFfclFh`mF+FhclF+#F;F\ijl/Fccl-F-6#,*F+F+F0F^vFjemF;Fj]oF5,$* (,&F;F+*$-F-6$F[i_o$"+i"o.@$FgdlF5F+F+,(F_i_oF5F`i_oF4*$F`i_oF4F1F;),(F4F+F`i_o F+F_i_oFboF]qF+F;FavQ,EIS~A052939F$"$I*,$*.,,FhpF+F*Fja_nF9!$]%FjxFjgalF[alFhim F+F[[mF;Ff]lF;FieoF;F]^pF;F_^pF;F;7#-F>6,Fg^p7%FB<#/FB-FV6'Fd^lFdfoF]_pFd[mF`_p Fis77FhpFcdmF`]lFf\`l"$z*Fhir"&:0#"&Do*"'zHY"(laB#")vU(3"")D;?`"*zLXh#"+0XT*G"" +N/vwj",D;q0;$"-z,(f#p:"-XNf[-y".&*\U/Q)Q"/D3-F&\$>"/z85J/Z'*Fji_o<(Fbh\o/FbpFg j_o/FioFf\`l/F_pF`]l/FeoFhp,.FeqF^b]lFfuFd`nFjzFbg`nF[rFb\lFjblF;FcelF+,,F^]mF+ F+F+F`hoF+FgcpF+FicpF+FicpF^dpQ,EIS~A001552F$Fa__l,$*(,(F5F+F*F[yF9F+F+,*F9F+Fj xF;F*FboF+F+F;Ff]lF;F;7#-F>6,Ffan7%FB<#/FB-FV6$-Fe^l6#FejhlFd^lFis77F5F5FdpFe]l FcdmF[tFc_sFdg\lFhbrF[]gn"%$y""%0S"%)**)"&<-#"&Ea%"'q?5"'[$H#"'R`^"(bz:""(+>g#" (:k%eF_\`o<'FibnFa[qF]am/FioFe]l,,FeqF;FfuF+FjzF5F[rF;F;F+,&F+F+-F^cl6$,$*&,(Fc clFboFfclF+F;F+F+FhclF+#F;Fe]l/Fccl-F-6#,*Fj]oF+FjemF;F0FboF+F+F+,$**,(F5F+-F-6 $Ff^`o$"+z'=/X%FgdlF[y*$Fj^`oF4F+F+,(F^_`oF^vFj^`oF5*$Fj^`oF5F5F;,&F;F+Fj^`oF+F ;),(F5F+Fj^`oF+F`_`oF;F]qF+F;FavQ,EIS~A052949F$Fadu*$FfglFabn7#-F>6,Q-Powers~of ~17F$7%FB<$Ff\al/FB-FL63F_sF_sF_sF_sF_sF_sF_sF_sF_sF_sF_sF_sF_sF_sF_sF_sF_sFgn7 7F+FabnFfddl"%8\"&@N)"(d)>9")pv8C"*t'Q.T"+Tuvvp"-(\wye="".\/!R*f,#"/LwI'*=FM"0h (HsBAEe"1Pf!H.yX!**"3H4Sfl#yPo"Ff\im"5"[oomv=>h'["6xTwOj)=ESs#)"845*\sn?X%3jS"" 9`r%[K^^oNC2R#":,;SADdZmSJU1%Fe_`o<$Fdo,&FfuF+FeqF^fzF^r,$*,F[vF\vF5FfrF]qF\v-F c[l6#*&,&F+F+-F_jlFdarF+F+F]qF+F+FigyF+FfrQ]oArrangement~of~n~distinguishable~o bjects~in~17distinguishable~urnsF$Q,EIS~A001026F$Fhjjl,$*&Ff]lF+,*F+F+F*F;FjxF[ yF9F5F;F;7#-F>6,Ffan7%FB<#/FB-Fe^l6#-FL6$FdghlFc`mFis77F+FinFdpF5Fd]mFd]mF[_qF_ iqFd]enFf^gn"%AA"%YV"&U<""&#oC"&eH'"'-#Q""'q1M"'ivw"(QQ&="(YFC%")uH75F\b`o<&Fdo F[pFa[q,*FeqF5FfuF[yFjzF;F[rF+-F^cl6$,$*&,(F;F+FcclFbqFfclFcdmF+FhclF+#F;F`_l/F ccl-F-6#,*F+F+F0F;FjemF[yFj]oF5,$*(,&F;F+-F-6$F`d`o$"+bD6,Ffan7%FB<#/FB-FVF]\anFis77F5F4F]blF\`oF^`nF_[qFdb]l" %(=#"%il"&$o>"&]!f"'Zr<"'U9`"(BVf""(qHy%")2*[V"")An/V"*j,9H""*!\?uQ"+n9Ei6"+-Wy '[$F_e`o<&Fd\xFibnFg_]n,*FeqF^vFfuFboFjzF+F5F+,&FgcpF+-F^cl6$F^cnF]cwF+FgcpFavQ ,EIS~A052929F$Fe[`l-F_jl6#,$*$,&F;F+F\dzF+F;F;7#-F>6,Fav7%FB<&Ff\alFb`yF`\n/FB- FIF^\nFgn77FinF+FdpF\_lF]jq"%$3""&p;""'70:"(YDE#")d>&)Q"*fZR](",a*=@5;"-sDPe+Q" .^YyHey*"0<$Q[miHF"1m$p&QGe*>)"3/Rr!e**>!RE"40h!R(y!)p)f!*"6>!4,"[Q[%o/L"8%>`(\ (*p#4eKw7"97)e*Q&yo@jVK?&F]g`oF^rF^r*&F]qF;)-F-6$,(*&F0F+-Fc[l6#-Fc[l6#,$F0F;F+ F+F`i`oF;F+F+$"+<8fgYFgdlFfjlF+FavQ,EIS~A009324F$F\\]l,$*&FieoF+,(F+F+F*F^vF9F5 F;F;7#-F>6,Ffan7%FB<#/FB-Fe^l6#-FL6$-FV6$F`[elFNFNFis77F+F+F4Fe]lF\_lFfdmFe_lF_ \[o"%N5"%FG"%Bx"&*4@"&Vw&"'$[d""'^-V"(na<""(N9@$"(.Qx)")v/(R#")b&)[l"*f!=*y"Fhi `o<%FdoF^u,*FeqFboFfuFboFjzF+F+F+-F^cl6$,$F_cnFj_w/Fccl-F-6#,(F+F+F0F^vFj]oF5,$ *(,&F;F+-F-6$Fbghn$"+QSDgOFgdlF5F+,&Fa\aoF4*$Fa\aoF4F1F;),&F5F+Fa\aoF5F]qF+F;Fa vQ,EIS~A052948F$"%25,$**,(F4F+F*FhzFjxFgimF+Ff]lF;FieoF;F]^pF;F;7#-F>6,Fg^p7%FB <#/FB-FV6%Fd^lFdfoF]_pFis77F4F2Fb\lF_blFabqFdcoF[]gn"%;B"%=o"&'>?"&u+'"''>z""'Q b`"(;Dg""(a$*z%")w;Q9")eA6V"*O7FH""*MEo(Q"+cdyi6"+yH$y[$F[]ao<&,*FeqF1FfuFhpFjz F;F5F+Fbiv/FeoF4F^fhl,(F+F+F`hoF+FgcpF+FgcpF^dpQ,EIS~A001550F$F`ben,$*&Ff]lF+,( F+F+F*F\cuFjxF5F;F;7#-F>6,Ffan7%FB<#/FB-Fe^l6#-FV6%FfevFNFNFis77F+FdpFd]mF^`nFi fgm"%1<"%#y("&)\N"'E>;"'M'Q("(=$pL")A$p`"")uz5q"*E7!)>$"+#=!ze9"+e%[Vl'",E>;a.$ "-9F6i%Q""-=(RAgJ'".iJ(p3")G"/ujQIA98Fj^ao<%FdoFhbw,(FeqF5FfuF\cuFjzF+-F^cl6$,$ Fd]anF]ebl/Fccl-F-6#,(F+F+F0F\cuFjemF5,$*(,&F`[lF+FfeblFg[lF+,(#F`crFdpF+Ffebl# F\cuFdp*$,&FajlF+FfeblFg[lF5F[yF;)FhaaoFfjlF+F;FavQ,EIS~A052913F$"$%*),$*(,(Fbo F+F*F5F9F+F+FdcmF;FieoF;F;7#-F>6,Ffan7%FB<#/FB-FVFbhamFis77F5F5FdpF:FcuF[tFcbnF gaoF_e_lFfbnF]h\nFf^\lFi[xF^h\nF_h\nF\\xF`h\nFa]\lF_\xFih[lFah\nF\bao<'Fibn,,Fe qFboFfuF;FjzF;F[rF+F+F+Fa[qF]am/FioF:,&F`hoF+-F^cl6$Fi[aoFjiamF+F`hoFavQ,EIS~A0 52935F$"$B*,$*&FdcmF+FdhbnF;F;7#-F>6,Ffan7%FB<#/FB-Fe^l6#-FL6$-FV6$F^dmFNFNFis7 7F+F+F5F4F2F]blFd]mF[tFgdmFjhmFg^mF\t"$p&"%75F^e`l"%,K"%$p&"&D,""&2!="&D?$"&cp& F_cao<(FdoF^uFdflFhoF[jbn/FbpF2-F^cl6$,$*&,,!%25F+Ffcl!$,*Fccl!%z=Fa]o"%(="Fdcl !#&*F+FhclF+FdjbnFfjbn,$*(,&F;F+Fc[cnF+F+Fb[cnF;Fe[cnF+F;FavQ,EIS~A052972F$F[fh m,$*(F*F+,&FhhdlF+F*FdpF+Ff]lFboF;7#-F>6,Fj]l7%FB<#/FB-FV6$F^[[lFainFis77FinFcd mF^blF[`nFeclFacoFidmFd]qFe`qFb_lF[[qF]jqFb`nFg^nFdg\lF\j\lFg^mFaboF^f[lF_^\mFe _ilFfeao<&F_u/F_pF^bl,(FeqF+F[yF+F]qFd[wFbh\o,&FdpF+F]qFgimFh[[lQ*~11~n~+~4F$Q, EIS~A017437F$F[gcn-F_jl6#,$*$,&F;F+Ff[fmF+F;F;7#-F>6,Fav7%FB<%FffalFifalF^d`mFg n77FinF+F4FgimF[\sF]cy"%1O"&\"Q"'c3Z"($zTm"*]')R0""+hITe=",'zfZ/O"-&QA$fEw"/UnJ )f"[<"0XEmY``J%"2ON)fl%R89""3&Q+%)>H(*)>K"451&)R,$Rd^'*"6$H[^/R(f#eS=2 p:i-"FhfaoF^rF^r*&F]qF;)-FeixF]pFfjlF+FavQ,EIS~A009444F$F]jq,$*$,(F;F+F*F5F[alF +F;F;7$-F>6,Ffan7%FB<#/FB-Fe^l6#-FV6%FNFNFjalFis77F+F5FdpFiqFabnF_blFeinFjin"$P $F`a`l"%'\""%_J"%Tm"&#*R""&![H"&7@'"'l38"'AdF"'C4e"(gRA""(&yyDFfhao<',(FeqF+F[r F5FjblF;FdoFa[qF]amFd\el-F^cl6$,$*&,*F\`oF+FdclFd]mFfclFjqFcclFiqF+FhclF+#F+F[_ q/Fccl-F-6#,(F;F+F0F5F_dlF+,$*&,&-F-6$F[[bo$"+whEYZFgdlFbo*$F_[boFdpF[yF;),&F5F +*$F_[boF4F+F]qF+F;FavQ,EIS~A008999F$"$z%-F>6,F`el7%FBF]iaoFgn77F+F5FiqFeclF\]x Fcjv"&?Z&"'+k!)")Sye8"*+[kd#"++[oGa"-+OvP"9++s9SR'4P()))["":++S#e'zb^?JRF'Ff hao<'FdoF[hoF]amFf_]n,(FfflF+*&,&F]qFboF7F+F+F[rF+F+FjblF+*&FbjaoF+F_glF+F\[boF avQ,EIS~A052667F$"$:',$*&Ff]lF+,*F+F+F*F^vF9FboF[alF5F;F;7#-F>6,Ffan7%FB<#/FB-F e^l6#-FL6$Fg[flFc`mFis77F+F5F2Ff`mF\]sF_ealFhg_n"%9>"%1g"&])="&e"f"'ec="'iEe"(- 'G="(1)Qd")E/,=")eJ_c"*#))*Qx""*')Grc&"+AT;Z<"+9eA$[&Fc]bo<'FdoFdfnF]am,*FeqF5F fuFboF[rF^vFjblF+Fjdx-F^cl6$,$*&,*!$"\F+Fccl!$.&Ffcl!$s#FdclFgjalF+FhclF+#F;"%f K/Fccl-F-6#,*F+F+F0F^vFj]oFboF_dlF5,$*(,&F;F+-F-6$F^`bo$"+F+Q'=$FgdlF+F+,(Fb`bo F4*$Fb`boF4F2*$Fb`boFdpF7F;),(F4F+*$Fb`boF5F5Fg`boFboF]qF+F;FavQ,EIS~A052958F$" %H5,$*(F_hpF;FdcmF;Fb]glF;F;7#-F>6,Fhcm7%FB<#/FB-FL6%FbipF^dmF[^glFis77F+FinF+F +F+F+F5F5F5F4F4F4FdpFdpFhpFhpF2F2Fe]lFe]lFiqF^abo<,FdoF[pFccclFgblFi\lFhblFcbzF fcclFbdcl,6,Ffan7%FB<#/FB-FVF]dmFis77F4F5F5F4F5F5F4F5F5F4F5F5F4F5F5F4F5F5F4F5F5 Ffbbo<&FdflFg^aoF]am,*FeqF;FfuF;FjzF;Fe]lF+-F^cl6$,$*&,(FfclF5FcclF4F5F+F+FhclF +Fj_wFjiam,(#Fe]lF4F+*&-Fc[l6#*(^#Fe`blF+F[vF+F]qF+F+,&F]vF+*&^#Fb^nF+F4FfrF+F; F+*&-Fc[l6#*(^#FbgyF+F[vF+F]qF+F+,&F]vF+*&^#F]vF+F4FfrF+F;F+FavQ,EIS~A052901F$" $y),$*&,(F;F+F*F+*$,*F+F+F*FboFjxF+F9F[yFfrF+F+F*F;F\v7#-F>6,Ffv7%FB<%Fa`hnFivF `_[lFgn77FinFinF5F2FcelFhyFc_flFic\mFdjv"(!)[V*"*+wp)="++sM^T"-+#R$[:5".+/6Nus# "/+?*\CK$z"1+SIAV])[#"2+?*4mk%zQ)"4+SUZU,47-$"6++Ci&*f;\v:""7+?z4QS,IJ-Z"9++K+I ca=n,>?,$*&F*F;,(F+F+F*F;F_eboF;F+Ffr<(FdflF_uF[pF]glFdz,**&,,F`qF\bvF]qFboFcqF [yFjqF+F\qF]a[oF+FeqF+F+*&Fa[_lF+FfuF+F+*&,(F]qFd[wF\qFboFhhdlF+F+FjzF+F+*&F\cl F+F[rF+F+F^r,$*,F[vF\v-F-6$,*F;F+F0F5FjemF;Fj]oFdp$"+.!3/K%FgdlF;,(*$F`gboF5Fbo F`gboF5*$F`gboF4FjqFfrF]qF]v),(F5F+F`gboF;FfgboFdpF]qF+F`[lFavQ,EIS~A052723F$F^ c\n,$*&F[[mF+,(F+F+F*F\cuFjxF+F;F;7#-F>6,Fav7%FB<#/FB-Fe^l6#-FL6$-Fe^l6#Fe^zFNF is77F+F+FdpF\_lFa_l"$O%"%*3#"&4+""&cz%"'r(H#"(**35""(CZF&")@FFD"*")))3@""*%or,e "+R&p(zF",6gn=L"",;0h8Q'"-plP\dI".HBFL\Y"".w])p"*=qF[hbo<%FdoF^u,(FeqF+FfuF\cuF jzF+-F^cl6$,$*&,&F;F+FcclF2F+FhclF+Fcael/Fccl-F-6#,(F+F+F0F\cuFjemF+,$*(,&F:F+* $Fc\nFfrFboF+,(F]^bmF+FijboFb\cl*$,&FehzF+FijboF\vF5FboF;)F\[coFfjlF+F;FavQ,EIS ~A004253F$"$A%,$*(FebxF+F*F+Ff]lF;F;7#-F>6,F^hl7%FB<%F^cxFcdwFc`yFgn77FinFinF5F cdmFcgq"%I5"&W1""'Gu7"(W4v"")'HDt#"*g")Gz%"++%ecN*"-+Su09?".++_XKu%"0+%enYM87"1 +?(yAw-N$"2+%e*Hnb4$**"4+GDE%Rf8XJ"6+WT`5zC^(f5"7+cqRS$3z0^y$"9+?$G6"yx=uNG9F`[ co<&FdflF_uF[p,**&,*FcqFboF`qFhzF\qF\bvF]qFhzF+FeqF+F+*&,*F`qFhpF\qFh`mF]qFedmF d]mF+F+FfuF+F+*&,(F]qF^fzFhhdlF+F\qF[yF+FjzF+F+*&F\[lF+F[rF+F+F^rF\exFavQ,EIS~A 052861F$Ffjbl,$*(F]iclF+FjfclF+F*F;F\v7#-F>6,Fav7%FB<%F`hxFb]zFbhxFis77F+F5F4Fe ]lF\_lFb[sFh[]lF\iv"%f="%#H'"&e;#Fgafl"')zm#"'74&*"(StT$")&GpB""):D0X"*gC+l""*! \$G2'"+!*=!\C#"+5OQJ$),$*(F*F;FijclF+FjfclF+Ffr<&Fdo,(FchcmF+*&,&FhpF+F]qF4F+Ff uF+F+F\]lF+FdhmF]amF^r,$F[h_lFajlFavQ,EIS~A005807F$Fijhl*&F*F5FfglFdp7#-F>6,F^h l7%FB<$Ff\alFc`vFgn77FinFinF5FcelF]w"%!G"F^wF^t"'w$H#"(['z6"(S#)*e")Se$)G"*K?TQ ""*C9Ja'"+7LX`I",S9'G49",S%4XUk"-Ghxd?H".wD**fUJ"".G4E:b(e"/!of6S8h#F^_co<&Fdfl F_uF[p,&Fj\mF+Fjj_mF+F^r,$*0F[vF\vF5FfrFccdlFdpFcgyF+F]qFb^nFj[yF+FigyF+Fe^alFa vQ,EIS~A052780F$"$P(*&F]^pF+FhiblF;7$-F>6,Fav7%FB<'Fca_lFaggmFbhxFi_al/F_s-Fe^l 6$Fe_alFjhlFis77F+FinF+F4F]blFbbnF``nFa_m"%*=""%FR"&qH""&PG%"'"[T""'!Gn%"(@La"" (Vs4&")]]$o"")$R-c&"*HAk$="*!3Hlg"+p%HK+#Fi`co<%FdoFf[clF[p-F^cl6$,$*&,&F^vF+Fc clFgimF+FhclF+F^h^lF\\cl*(,&#Fd[wF5F+Ffh^lFb^nF+Fc\clF;Ff\clF+FavQ,EIS~A006190F $Fdg\l-F>6,F`el7%FB<#/FB-Fe^l6#-FL6%FNFNF]_pFgn77F+FinF5Fd]mFhyF]]x"&![yFia`m") ![Sz%"+g(H]U"",+g`lq%".+;'f"*4<"/+'p`ipx'"1+SUzAw4H"3+_2`v3WX8"4+SUn&=S`lm"6++S uR!R@OAN"8+!GP.`@;sqx>":+?r()*z3@yhuv6";++cMo\wu$3H"yt"=++;/a?C(oz*4mt[Fi`co<%F i_anFdoF[p*&FabcoF+F_glF+FfbcoFavQ,EIS~A052635F$"$"e*&F*F2Ff]lFbo7#-F>6,Fch_l7% FB<$Ffh_l/F_s-Fe^lF[gamFgn77FinFinFinFinFinFinFhgnFg`vFfablFgablFa[amFb[am"++7, `LFf]dlFg]dl"/++oVn28Fidfl"1+?:P"\#oU"2+SYu"e3B$)"4+![Os09..<"5++g\E7IN\OFfdco< *F_uF[pF^pFapFfavFepFgav,&FfeilF+*&,&F]qF+F\cuF+F+FfuF+F+*&FhecoF+F_glF+F]qQMPa irs~of~sequences~of~cardinality~at~least~3F$Q,EIS~A052521F$Ffin,&-Fc[l6#-F^cl6$ *&,&-Fc[l6#-F^cl6$,$*(F\j`mF+,&F\j`mF+F;F+F;Fd_xF;F;F[`xF+F;F+F+FjdpF;FbepF+F;F +7#-F>6,Fav7%FB<&/F_\nFd^l/F_s-FL6$FNF_\nF`gmFc]hlFis77FinF+F4F2Fb\lF\`oF[\sF]f qF\`l"$C%F\_`l"%F:"%qG"%z_"%5(*"&Aw""&x=$"&+r&"'()=5"'1/="'1"=$F\fcoF^rF^rF^rFa vQ,EIS~A001970F$Fiin,$*,Ff]lF;FdcmF;FeazF;Fb]glF;FgbelF;F;7#-F>6,Fhcm7%FB<#/FB- FL6'Fd^lF^dmF^bzF[^glFccelFis77F+F+F+F5F5F4FdpFhpF2FiqF]blFgimFb\lFcuF\_lFcdoF^ blFi`mFddmF[amFedmF`hco<7FdoF^uFgblFjd`lFac[l/F`arF^bl/FcarFi`m/FfarFddm/FiarF[ am/F]arFcdoFa^gl/Fa`rFgim/Fd`rFb\lFb^glFc^glFd^glFeh[nFfh[nFe^glFf^gl,V"'C]DF+F ]q"&]a%FeqFjflFfuFbi]nFjzF_e`lFcqF+Fe\oFfi]nF[r!$c%Fjbl!$W(Fb`o!%S9F\q"%NIF`qFb ^mFahsF_e`lFehsFjflFihsFbi]nF]isFfi]nFaisFficoFeisFgicoF^^t!%k?Fb^t!%w6,Ffan7%FB<#/FB-Fe^l6#-FV6&FNFNFNFjalFis77F+F4F:F\`oF^`nFe_v"$c( "%&H#"%np"&]6#"&1U'"'8\>"'16,F``x7%FB<'Fc`x/%#T4GFN/F[ax-FL6$FN-FF6#FfadoF\axFaaxFis77FinF+F+ F5FdpF:F\_lFc\lFh_qFaboF]hbl"$Z)Fgcbl"%nO"%kv"&kb""&^=$"&()\'"'J?8"'ruE"'\*R&Fe _doF^rF^rF^rQ=Unlabelled~trees~of~height~4F$Q,EIS~A001384F$Fhdm,,F^_coF+FgdyF[y FhdyF2FidyF[yFjxF+7#-F>6,F^hl7%FB<$F_eyFc`vFgn77FinFinFinFinFinFinFhgnFg`v">) Fidil"(gTn$")+G^?"*SY/3""*+!)oX&"+oj.rE",+7&Rw7",+7Xc)f"-+;k*\w#".W*G6ph7".?jZ? "*p&"/+Kq>^_DFhbdo<*F_uF[pF^pFapFfavFepFgav,,*&,,F[h[oF+F]qFjamFcqFcelF`qFhyF\q FdgmF+FeqF+F+*&,,F`q!$+%FjamF+FcqFiflF]qF_`nF\q!$])F+FfuF+F+*&,*F]q!$?%F`qF^[qF cqFddmF\qFh_lF+FjzF+F+*&,*FcqFgdlF]qF[amF\qF]blF`q!#SF+F[rF+F+F]cvF+F^rFe`coFav Q,EIS~A052792F$"$\(,$*&Ff]lF+,,F+F+F*F;F9FboF[alF5FjxF;F;F;7$-F>6,Ffan7%FB<#/FB -Fe^l6#-FL6%FNFN-FVFjfblFis77F+FinF+F4F5Fe]lFddoFd]mFj[oFc_sF_iqFiejlFeien"$i(" %<9"%VD"%UY"%&\)"&*Q:"G"&x6&F[edo<'FdoFhoF[pFgbl,,FeqF5FfuFboFjzF;F[rF;FjblF +-F^cl6$,$*&,*Fe_mF+Fccl!$1"FfclFbbnFdclFh`mF+FhclF+#F;Fjhcl/Fccl-F-6#,,F+F+F0F ;Fj]oFboF_dlF5FjemF;,$*(,&F;F+-F-6$F^gdo$"+$**Q8\&FgdlF+F+,*FbgdoF+*$FbgdoF5F5* $FbgdoF4F2*$FbgdoFdpF7F;),*F+F+FbgdoF+FggdoF5FhgdoFboF]qF+F;FavQ,EIS~A052546F$F ihdn-F>6,F`el7%FBFbedoFgn77F+FinF5Fd]mFeclFdgmFi`gmFdebm"(?Jl"Fdiz"*+OrU%"++_6S &*"-+Onf;?".+'\)*)\u%"0+/jQ;`B""1+S#y"fTDL"2+g4g1fBr*"4++_v;qk:-$"5+?>[u&*Gh_)* "7+SA?3o4x.;M"9++G0fXsgi3X7F[edo<'FdoFdflF_^dlF[p,,*&,,FcqF5F`qFf`mF\qFidmF]qF_ `nFeclF+F+FeqF+F+*&,*F`qFboF\qFe_mF]q!#_F_fuF+F+FfuF+F+*&,(F]qFa`lFhzF+F\qF;F+F jzF+F+F[glF+FjblF+*&FdfdoF+F_glF+F_gdoFavQ,EIS~A052681F$F]gp,$*(F*F+FgfzF+Ff]lF ;F47#-F>6,F`el7%FB<#/FB-FL6$F`gzF__pFgn77FinF4FjqFd]mFa^mFa_mFdelF]g[lFfabl"(S' )3"FihemFhen"++[+P9",+C1"o="-+O([`h#".+S5BI#R"/+Smp$oF'"1+!)G%Giq1""2+S=<672#>" 3+g\E7IN\O"4++#*HX-1()H(Fhido<'F_^dlF_uFahnFd\xFcjw,$F_glF4F4FavQ,EIS~A052637F$ F]\gn,&F+F+*$,&F4F+FfglFboFfrF;7#-F>6,Fav7%FB<%F_eyFb]z/FbsFa]_nFgn77FinF+F5Fe] lFj[oF_^v"%(z$"&d6&"'cj")")"e]]""*".T&))["**Q&"0ih4OR)f;"1 2LaYjM@b"3h%=HRtTH(>"4E#[N\hC*o`("6d!z,H$eA/_1$"8d]g;TqaCUAK""9w3;!47P?gh+.'F`[ eoF^rF^r,$*,F[vF\vF4Ffr-F_jl6#Fb^nFfrF]qF]v),&FhgyF+FjdnF;FfjlF+FfrFavQ,EIS~A00 6677F$Fjin*(Ff]lFboF_hpF;FeazF;7#-F>6,Fhcm7%FB<#/FB-FL6&Fd^lFd^lFbipF^bzFis77F+ F5FdpF2F:FddoFd]mFcelF]_lF]`oFcirFihpF]`nFbcoFb_lFjhqFd^rFg_lFi_lF_^\mFcfjlF`]e o<(FdoFdzFa[qF]amFiir,6FeqF2FfuFjqFjzFjqF[rFjqFjblFjqFe\oF2F]q!$Y"!$O$F+F`qF;F\ qF[bv,.F`q#F+FihpF\q#F:F[amF]q#FejrFihpF[chmF+-F^cl6$,$F_\blFb^flF^czF+F``uF`v, $F`qF_^eoQennumber~of~ways~to~make~n~cents~with~coins~of~1~1~2~5~centsF$Q,EIS~A 001304F$Fh^m,$*(FieoF+Ff]lF;F]^pF;Fbo7$-F>6,Ffan7%FB<#/FB-FV6$Fd^lF]_pFis77F5Fd pF]blFh`mF^`nF`jnFdb]l"%)=#Fje`o"&%o>F\f`o"'[r6,F`el7%FBF^_eoFgn77F5FdpFf`mF`j\l"%o>Fh`gl"'+c_")?v-6"*S)zXE"+?* HH9("-++k!G9#".+kG"=rq"0+sIo:ca#"1+#R*4())y#**"3+S'3Y^6(pT"5+S9!R+#*pj(="6+g8ZJ x>vl+*"8+SuZc0(zKN$f%":++smvDO#e2T![#"<+gP&[![wh(HT$QT""=++GpF"[y/Px/I[)Fh^eo<% Fibn,(*&,(F\qF4F]qF:F2F+F+FeqF+F+F^_imF+FjzF+Fhbw*&F\`eoF+F_glF+FgcpFavQ,EIS~A0 52573F$Fa[\o,(F_emF+F*Fj_w*$,(F+F+F*F7FjxFdpFfrF\i^l7#-F>6,Fav7%FB<&/F_s-FVFd]e l/FBF^egl/Fbs-FL6$FBF_\n/F_\nF\gmFis77FinF+F+FdpF\_lF_`nFejz"%/L"&r+#"''*\7"'uP z"(K17&")-JYL"*3+1@#"+3.$QZ""+#>'=/**","R,a,n"-'RnE>c%".MHAG?7$"/k)*\pxY@"0=5M) fY#["Fiaeo<&F^uF_uFgbl,(F[^[lF+*&,&F[yF+F]qF7F+FfuF+F+FhhmF+F^r,$**F[vF\v,(FiqF +Fhd^nF[y*$,&F+F+Fhd^nF\vF5F7FfrF]qF]v)F_deoFfjlF+F^fdlFavQ,EIS~A007564F$"$V%-F ^cl6$*(FddpF+FjdpF;-F_jl6#,$*$,&F;F+FaepF4F;F;F+Fbep7#-F>6,Q7Necklaces~of~3~col oursF$7%FB<&FgfpFhfp/FB-FI6#-FV6%F`fpFafpFbfpFifpFis77FinF4F2FgimFcelFfdmFb[rFd brFce`m"%&>#"%Mf"&2h""&oV%"'VE7"'-=M"'Nm&*"(W3p#"($['f(")UX_@")f;6,F^hl7%FB<%Fc_^nFc`yF_ eyFgn77FinFinF5FjqFe[qFc`an"%#o%"&#HZ"'Mea"(gs3("*ivC-""+sDjA;",%fn:4G"-!Q[$eo_ "/U/(HMT1""0_y(4>$GI#"1a8)>oacJ&"3+f8Hqwq.8"4A`%oDjY`&Q$"5K6TG$>te,G*"79J?%QWiz (oxEFdfeoF^rF^r,$Fjc_nFfrFavQ,EIS~A052875F$Fec^l**Ff]lF;FdcmF;FeazF;Fb]glF;7#-F >6,Fhcm7%FB<#/FB-FL6&Fd^lF^dmF^bzF[^glFis77F+F+F+F5F5F4FdpFhpF2Fe]lF:F]blFjqFb\ lFcuF\_lFc\nFcelF\`oFi`mFj`mF]heo6,Q4Schroeder's~problemF$7%FB<#/FB-FV6$FN-Fe^lF^hcnFis77FinF+F+F4FgimFedmFd`e l"$.*"%zU"&$z?"'\I5"'f)=&"(Bnk#")p)[O"")t$R5("*>Nps$"+>:!)o>",`$yNY5",4I,4f&"-j pUf,I".(ee@O=;F^jeo<&F^uF_u,(*&F^hoF+FeqF+F+Fj^elF+FhhmF+FgblF^r,$F]_elFbrQdoSc hroeder's~problem.~Dissections~of~a~polygon~or~parenthesizing~a~productF$Q,EIS~ A001003F$Fc\l-F>6,Q-Planar~treesF$7%FB<$Fcs/F_sFhjeoFgn77FinF+F5Fd]mFfi[mF[[[o" 'S=9"(?6b%"*!GHD<"+SQOXv"-+7@WRP"/+7$4>62#"1+ov^Xyn7"2+_Zf6z"*\)"4+wTfg9"4$>'"6 +?*4-;>wht["8+?()RA@80#G>T":+!))eg\&fIFj+g@PT95;dJO#H^O"@ ++o2))QMUR$eXlJPRF^jeo<&F^uFdflF_uFg^elF^rF[\foQIPlanar~labelled~trees~with~no~ unary~nodeF$Q,EIS~A032037F$Fe`q*&,(FboF+F[alF+F9F;F+Ff]lF;7#-F>6,F`el7%FB<#/FB- FV6%Fd^lFd^lF`dmFgn77F5F5FdpFd]mFeclFhyFaiwFg`vFdg[lF]^`lF^^`lF_^`lF`^`lFff_lFa ^`lFb^`lFc^`lFd^`lFe^`lFf^`lFcgdlFg]fo<(F_^dlFahnFibnFa[qF]amFegdlFfgdlF5FavQ,E IS~A052628F$"$u&*&,(F*F;F`adlF;*&F`adlF+F*F+F+F+Ff]lF;7#-F>6,F^hl7%FB<%F[gmFcdw F^adnFgn77F+F5FhpF\_lFjjp"$@'"%rZ"&tE%"'tYV"(L%f\")"*)pD'"*T**)R')",LQ;\H""-8?U .#4#"..c"e>AO"/@\bx9)o'"1@zb3qN68"2`BMzQI-s#"3V=8@?%e.&f"5t,R(*Rz=go8"6TWHiU*)H 38I$Fe^fo<&FdoF]amFidx,*FdezF+*&,(F\qF4F]qF]blFiqF+F+FfuF+F+*&,&F]qF^vFa`lF+F+F jzF+F+F[rF+F^rF^rFavQ,EIS~A052866F$"$N),$*&,&F+F+F*Fe]lF+Ff]lF^vF;7#-F>6,Fhgv7% FB<&F[hvF^hvFdhv/FB-FV6*F_sF_\nF_\nF_\nF_\nF_\nF_\nF_\nFis77F+F]blF\`oFdinFf^qF d^mFh_lF_jnFc^\mF`dpFi[\nFc`an"$P'"$U("$b)"$w*"%06"%U7"%(Q""%S:Fd`clFf`fo<%FdoF i`n,*FeqF;F\qFdpF+F+F]qFhp,(F\qFdpF+F+F]qFhp,$F\qFdpQ1~7-gonal~numbersF$Q,EIS~A 001107F$Fa^v,$*(FgfzF+Fh\zF;Ff]lF;F;7#-F>6,Ffan7%FB<#/FB-FL6$Fi]`mFg[glFis77F+F 4FhpF:FcdmF`crFj[oF__lF``nFj[vFd_vFjhfmFg^al"%>7"%t>"%$>$"%n^"%h$)"&HN""&"*=#"& @a$F`bfo<&FdoFd\xFidxF_fgl,&F;F+-F^cl6$,$Fd]anFddyF[\_lF+,$**,(Fa\_lF;F;F+Ff\_l F+F+Fe\_lF;F`\_lF;Fg\_lF+F;FavQ,EIS~A001595F$"%>5,$*(FieoF+Ff]lF+,*F;F+F*FhpFjx F1F9F+F;F;7#-F>6,Ffan7%FB<#/FB-Fe^l6#-FL6$-FV6$F`[elFd^lFNFis77F+F5F2F\_lF[\oFd `elFe`elFg`elFi`elF[aelF]ael"'D(H#"'*)eu"(]=U#"(TO'y")%*H`D")u\!H)"*Z0>p#"*&)e0 u)"+<6/QG"+A31:#*F]dfo<&FdoFdfnF]am,*FeqF+FfuF1FjzFhpF[rF;-F^cl6$,$*&,(F5F+Fccl F^vFfclF+F+FhclF+Fcael/Fccl-F-6#,*F;F+F0FhpFjemF1Fj]oF+,$**,&F;F+-F-6$Faffo$"+% G&yzIFgdlF5F+,&F;F+FeffoF+F+,(*$FeffoF5FjqFeffoF\cu*$FeffoF4F^vF;),(FhpF+FeffoF 1F[gfoF+F]qF+F;FavQ,EIS~A052975F$"%Z5*&Fd`_lF+F*F+7#-F>6,F^hl7%FB<%Fa`hnFjg^nFb `hnFgn77FinF+F5FcdmF[jn"%bJ"&;&p"($er=")k?af"+>y\&=#",+h()44*".$)*z.IEU"0W:=8?: <#"22rA+I*)>A""3/%4$*oa0WZ("5b]#RZP\"73]dH3KtL=.N"9jN5?6@qs*Qpl#";[d_d'e<_p k6^9#"=zO43FO]m7(*\$o$="??T_n2yLe4!y%>ti;FagfoF^rF^rF^rFavQ,EIS~A053492F$F[b[o, $*&,,F+F+F*F[yFge_lF;FjxF5Fie_lF5F+F*F+Ffr7#-F>6,Ffv7%FB<%F^sFa]el/FB-FL6%F_sF_ sFNFgn77FinFinFinFinFinFcenFc_flF\bm"(g`$>F[eil"++Onv?",+OV8*z".+_J*\(Q$"0+_.iN )o:"1+!Ghl/W)y"3+!oVrmtWF%"5++KsC(\mp[#"7++kK_W8^!ea""9++W*e!f&GvHB-"":++S-\C(G i)y%or"<++!ob[@gJD,A7`,$*&,,F\vF+F*F+Fge_lFfrFe[]nF5FjxF+F+F*F+F;<*F_uF[pF^pFap FfavFcfy,(*&,*F]qF\_blF\qFdpFhddoF+F`qFiqF+FeqF+F+*&,(F\qF1FiqF+F]qF7F+FfuF+F+F jayF+/Fi`oFc_flF^r,$F[h_lFe^alFavQ,EIS~A052721F$"$x'*&F*F+FfglFdp7#-F>6,Fav7%FB <$Ff\alF]jxFgn77FinF+FiqFeclF_e_lFe_coFdb]n"&s'GFa]\l"'C)*e"(S9i#")OV`6")[;L]"* 3Q5=#"*'4C&R*"+S=`ES",%=p)zr"",KSW9I("-7`kP#4$".%)z0qcI"".!))Q"ev\&F^[go<%F^uF_ u,&Fj\mF+F]cmF+F^r,$*0F[vF\vF5Ffr-Fc[l6#Fg[lFdpFg[yF+F]qFfrFj[yF+FigyF+FbrFavQ, EIS~A002697F$Fa]v,$*(Ff]lF;FdcmF;FeazF;F;7#-F>6,Fhcm7%FB<#/FB-FL6%Fd^lF^dmF^bzF is77F+F+F+F5F5F4FdpFdpFhpF2Fe]lFiqF:F]blFgimFddoFb\lFcdmFabnFd]mFf`mFi\go<)FdoF ^uFgblFjd`lFac[lFf^gl,6FeqFboFfuF[yFjzF1F[rF1FjblF1Fe\oF[yFb`oFboF\qF+F]qFcdmFb [sF+,,F\q#F+Fi`m#F^blFedmF+F]q#F4F]bl-F^cl6$,$*&Fc[^nF+FhclF+F]czF^czF+F`fdlF+, $F\qFg]goQYnumber~of~ways~to~make~n~cents~with~coins~of~1~3~5~centsF$Q,EIS~A008 672F$F[a\m,$Fi_imF;7#-F>6,Fj]l7%FB<#/FB-FV6$Fc^zF_camFis77FinFe]lFjqFabnFg`mF\` oF[tF[`nFc\lFjhrFdinFgdmF\]sF__lFa^mFjdmF^`nFbcoFe`qFjjpFccqFa^go<&F_u,(FeqF+Fb oF+F]qF\cuFcjwF`jdl,&F5F+F]qFhpF]_zQ)~5~n~+~2F$Q,EIS~A016873F$"$U*,$*&FfglF;Fbc bmF;F;7#-F>6,F^hl7%FB<%FahlF^sFghlFgn77F+FinF5F2F`rF\_m"%UB"&YO#"'=HF"(IOa$")#y B6&"*'G;8")",)z$yXS""-!>u"HME".A_[r1K&"0E*)[&fT^6"1y1*4MFyl#"2]zc9NQ&=l"4iEUG;L nFp""5mb?k'fOz+k%"7e:5#>A")*Q%)Q8Fa_goF^rF^r,$Fjc_nF`[lFavQ,EIS~A052841F$"$3),$ *$,(F;F+F*F4F9F+F;F;7$-F>6,Ffan7%FB<#/FB-Fe^l6#-FV6&FNFNFNF`dmFis77F+F4F:Fh`mFb coF\_m"$Q)F^ijn"%t!)"&d]#"&sx("'*QT#"'C#\("(WaK#"(@x@(")(Q-C#")0E`p"*Ob"e@"*&** [)p'"+!fz!z?"+1V0`kF[ago<&Fdo,(FeqF+FjzF4F[rF;Fd\xFahz-F^cl6$,$FgiamFfbz/Fccl-F -6#,(F;F+F0F4Fj]oF+,$*&,&-F-6$F_cgo$"+YN&=A$FgdlF^v*$FccgoF4F^vF;),&F4F+*$Fccgo F5F+F]qF+F;FavQ,EIS~A052541F$"$v%-F>6,F`el7%FBFbagoFgn77F+F4Fd]mF`j\l"%)3#"&+C$ "'gLg")S!4J""*gL]D$"+gTo#4*"-+O.>AG".+_Vwaj*"0+%eZ\z)e$"2+_Bd")e![9"3+_NQJeG#H' "5+gT;>EF]HH"7++C)H!y%3;[X""8+gX*p**H%H(Gww":++OVlK'R#fB')G%"<++))QpE.N[X)4HD"> ++%=hgRJS*z(['*p:F[ago<&FdoFd\x,(Fb[wF+FbfhlF+F[rF+Fbjz*&FibgoF+F_glF+F`cgoFavQ ,EIS~A052668F$"$;',$Ff]jmF;7#-F>6,F^hl7%FB<%Fc`yF_eyFagbmFgn77FinF+F4FcuF_\r"%@ :"&%=A"'"o$R"(.QB)"*=FM)>"+b643a"-(RV!eY;".#ep^DPb"0"HP$G0%Q?"1$G-,'H6`")"3W$Q\ \@$z?N"5,9y"f"yhfK;"6PU(>3i@Tt!4)"8?()RK(Q5Cq[nU":42&GZ")\CiO'pQ#"<"fB$Q'[J)*G& 4J7T"FgegoF^rF^r,$*.F[vF\vF5FfrFg[yFfrFd_jmFfrF]qF]vFh_jmF+FfrFavQ,EIS~A048802F $F`iv,$*$FaehnF;F;7$-F>6,Ffan7%FB<#/FB-Fe^l6#-FV6%FNFNF^bnFis77F+F5F2FcuF]hrFce nF[galFj^o"%[C"%)o'"&s#=F[h[l"'%QO""'3EP"(%)z,""(%="y#"(O$)f(")S!f2#")_Zrc"*%eZ \:"*sYKB%Faggo<%FdoF[ghnF]am-F^cl6$,$FeczF_emF_ghn,$*&FeghnF;FhghnF+F;FavQ,EIS~ A002605F$"$w%-F>6,F`el7%FBFgggoFgn77F+F5FjqFhaq"%c5"&+W""'ghB"(Se^%")gLq)*"+S9% pU#",+OV0j'".+glYE*>"/+W@a"G`'"1+kCmxB?B"2+3%*)e0hu))"4+?r"\HI)oj$"6+!ojUEwQy*e ""7++%y)>MZ&HPQ("9+gX*\%oJp.4JO";+!))=mIfy8Wh[)="=++3igW<&>YW2*H5Faggo<%Fdo,(Fb awF+F_cdlF+FjzF+F]am*&F[igoF+F_glF+F^igoFavQ,EIS~A052611F$"$c&,(-F-6#,*F0F;Fjem F+Fj]oF+F*F+F+*$F][hoF5F;F*F;7$-F>6,F[s7%FB<%Fc`yFddwFii\oFis77FinFinFinF+F4Fjq FcirF^[q"$F*Ffien"&0$>"&&G!*"'qvU"(Cj/#"(i=))*")C34["*L">cB"+!ed7;""+:]O`d",qK4 Q'G"-&3)R[J9F\[ho<(F_uF[pF^pFi\l,**&,(Fi`mF+F\qF\\oF]q!$N"F+FeqF+F+*&,(!$I"F+F] q!$2"F\qFhjpF+FfuF+F+*&,(F]q!$"G!$)>F+F\q!#"*F+FjzF+F+*&,(F\qFcdmF]qFj[qFb^mF+F +F[rF+F+Fc\oF^r,$**F[vF\v-F-6#,&FjemF\\lF\cuF+F+F]qF]v-Fc[l6#*&,&Fg]gmF+FdjlF;F +F]qF+F+F\i^lFavQ,EIS~A052703F$F]fhn-F>6,Ffv7%FB<%Fc`yFddwF_d^oFgn77FinFinFinF2 Fa^mFaiwFic\m"(+%e5")SmPP"+!3I3_"",+S)R0q".+!)G))Qg$"0+?69n![?"2+#R:6-Du7"3+W@M I%Q[h)"5+?B*z!)y8()G'"7+S5FsWIT";++#fI!Q"\"*G>No$"=++kgx%fkY 8t$o$[$"?++SMxr(el#o^-m#[$F\[ho<(F_uF[pF^pFdz/FbpFa^m,**&,(F`qF\\oF]q!$0"Fi`mF+ F+FeqF+F+Fh\hoF+*&,&F]qF`]ho!#**F+F+FjzF+F+F[rFcdmF^r,$**F[vF\vFe]hoF+F]qF]v-Fc [l6#,$*&,&Fg]gmF;FdjlF+F+F]qF+F;F+F\i^lFavQ,EIS~A052730F$FeiinFfho7#-F>6,Fhio7% FB<)F[joFajoFejoFhjoF\[pF`]hmF`[pFgn77FinF+F2FfdmF\iv"%cv"''>?""(c=?#")mEfX"+E5 &>0"",1`xSn#"-1>0p?u"/kQtj0JA"0YDun'3@s"2djy?y=C]#"3q$))eEh9;C*"5#[4$QcG"yBi$"7 9:)QW;?1_:]""8aSqnvk;i!ehl":Cgj3&y/8&=sT,$"6,F`el7%FB<#/FB -Fe^l6#-FL6%FNFNF`f`mFgn77F+FinF5F2Fa^mFhgnFg`v"'Si:"(+C#G")g@(p&"++k'4G"",+C-a ;$"-+7&3e`)"/+KG"*H$\#"0+k#f3VVy"2+!)[B]%fVE"3++Si1th)=6,Q0Multiples~of~12F$7%FB<#/FBF`^lFis77FinFjqFcelF_bl FeclFihpFa^mF[emFhaqFc^mFcenFd\lFe^mF\[qF`j\lFg^mF]wFchdlFi^mF]bhlFhyF_dho<%F_u Fddv,&FeqF+F]qFhzFe`lFe`lFcdhoQ,EIS~A008594F$F__m*$FfglF47#-F>6,Q,Powers~of~3F$ 7%FB<$Ff\alFggelFgn77F+F4F:F\`oFd]qF_[qFjbglFie`o"%hlF[f`o"&\!fF]f`o"'T9`F_f`oF ^\imFaf`oFidelFcf`o"**[?uQFef`o"+,Wy'[$F[eho<$Fdo,&FfuF+FeqF^vF^r,$*,F[vF\vF5Ff rF]qF\vFdgyF+FigyF+FfrQ\oArrangement~of~n~distinguishable~objects~in~3distingui shable~urnsF$Q,EIS~A000244F$Fdj[m*&Ff]lF;F]^pF;7$-F>6,F^ial7%FB<#/FB-FLFa_eoFis 77F+FdpFddoF[amFjhqF\\v"%$4"F\jem"%T)*"&C&H"&t&))"'?dE"'hrz"(%["R#"(`W<(")gL_@" )"3qX'"*W-r$>"*L28"e"++ARV<"+,m6,F`el7%FBFcfhoFgn77F+FdpF^blFhy"%/HFeccm"'gpy")+7`;"*?"* y'R",?"pOr5"-+Cq89K"/+g4#p11""0+wX%RT=Q"2+s'3h?=*[""3+OrMGbcai"5++[O_=Yb9G"7+!G 499yB')4N""8+SU:'3U[)H+*o":+SiQ*Q'\X7;1s$"<++S5*>X(4A>^27#">++k+G?t8pbr]CF"F^fh o<%FdoFdaeoFhbw*&FfghoF+F_glF+FhghoFavQ,EIS~A052577F$F\fx,$*&,(F*F+FafmF;FefmF+ F+Ff]lF;F;7#-F>6,F`im7%FB<#/FB-FF6$F_gmFa[zFgn77FinFinF+FdpFd]mFhaqFhf[lFcjvFic \mFifo"(?fE$FhelF[eanF[b`m",+/F^4)".+o2'\?7"/++_:^h>Fc[hm"1+?jxioYg"3+S5.nsU_6" 4+!3yw2pD6BFaiho<&F_uF[pFgbl,&*&F\cmF+FfuF+F+FhahmF+F^rF+QEPermutation~with~at~ least~two~cyclesF$Q,EIS~A001563F$Fi`m,$Fc]mFc\n7#-F>6,Q0Multiples~of~21F$7%FB<# /FB-FL6%-FV67FNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFNFd^lFd^lFis77FinFc\nFc\l FhdmF[emFadqFd^mFg^nF`j\lFdfwF^[qFghvF[_mF[^vFi]_lFdbrFgicl"$d$Fihv"$*RFf\cmFij ho<%F_uFfehm,&FeqF+F]qF[bv,$F]qFc\nFj[ioF][ioQ,EIS~A008603F$"$L$,**&FcioF+FfglF +F+FfglF;FcioF;F+F+7#-F>6,F^hl7%FB<%Fc`yF_eyF`gmFgn77FinFinF5F:F_blFihy"$t'Fa\i n"&1q""&F[*"'%fi&"(E]`$")R3VB"*%)[Dj""+A#f?>""+,O=(4*",cws%Qs"-aj$>@*f".(*))Rl1 :&"/9..;9*e%"0y8#zlXJUF]\ioF^rF^rFinFavQ,EIS~A052834F$Ffc^l-F_jl6#,$*$,&F;F+Fef mF+F;F;7#-F>6,F^hl7%FB<%FffalFb`yFifalFgn77FinFinF5F4Ff`mFb^mF]j[oFfjr"&/Z&"'31 e"(SS@(")!)G$p*"+7pR_9",!G@w]B"-39?-TT".g(*[&=Fy"0?Jh+yve""1S9mTk**HM"2)[j_J;fx y"4++K3MSFY">"5?:Lo8E[n7\Fd]ioF^rF^rF^rQ8Cycles~of~rooted~CyclesF$Q,EIS~A052804 F$F]gal-Fc[l6#,(F*F+FjxFfrF[alF`[l7#-F>6,F_fil7%FB<#/FB-FF6#-FV6%Fgfil-FIFfcem- FI6$FNFc`jnFgn77F+F+F5FdpFcuFb[sF_e_lFf]^o"%Ci"&;O$"'c'=#"(cmK""(KO*)*")Cm=q"*O v,u&"+wp/aW",w`#R2S"-KOtlrM"./>,9/P$"/w<@6kUJ"0;SdzSXG$F^_io<'FdoF^uFdfl,*Fb[wF +F\]lF+FjblF+F[rF;F``oF^r,$*4F[vF\vF5Ffr-Fc[l6#FgrF+FdpFfrF]qF\v-Fc[l6#*$F]qF`[ lF+-Fc[l6#,$FeffmFfrF+-Fc[l6#,$F]qF`[lF+)F]q,$F]qFg[lF+FbrQ]ppermutations~Sigma ~such~that~Sigma^4=Id,~degree~n~permutations~of~order~dividing~4F$Q,EIS~A001472 F$F`cr,$*(F*F;Fd[imF+,&F+F+Fd[imF+F;F;7#-F>6,F`[cm7%FB<&/%*ConnectedG-FV6$-FI6# %%TreeG-FL6$%*UndefinedGF]cio/FB-FF6#Fhbio/F`cioFd]cl/F]cio-FL6$FN-FFF\cioFgn77 F+F5F:FgclFbeclF\[[oF^hbnFcielFidel"+++++5",,YUPf#"-)oq$3IuFab]o"0WTDtZr$z"2D1R Dg#H>HF`jelF\a`o"7o>`'pQ(f:"f=#"9TG`T7"zH]t7/"":+++++++++!)GC&"<,k>P'[:&pWH%=#y #F_bioF^rF^r,$*.F[vF\vFh[yF]vF5FfrFg[yF\vF]qF\vFec`mF+FfrQ6Partial~maps,~(n+1)^ nF$Q,EIS~A000169F$F__l,$*(F*F+Ff]lF+Fb\`oF;Fbo7#-F>6,Ffan7%FB<#/FB-FL6$F[]`oFc` mFis77FinF5F5F2FjqFh`mF\]sF_^rFaam"$1("%'e""%kNFic_l"&%*z""&K/%"&]3*"'QT?"'%pe% "(w1.""(3fJ#"()z._Fgdio<&FdflF_uF]am,*FeqF+FfuF;FjzFboF[rF+-F^cl6$,$*&,(FcclF;F fclF+F+F+F+FhclF+#F5Fe]lFc^`o,$**Fj^`oF+Fa_`oF+F__`oF;Fb_`oF+FboFavQ,EIS~A05299 4F$"%r5,$*&,&F+F+F*F5F+Ff]lF^vF;7#-F>6,Fhgv7%FB<&F[hvF^hvFdhv/FB-FV6%F_sF_\nF_\ nFis77F+FhpFjqFg`mFddmFfdmFidmFe`qFhgqFc`nFc[[lF^[qF^`lFd_vFh_mFhegl"$D%Fjfdm"$ K&Fhbcm"$^'Fifio<%FdoF[gx,*FeqFboF\qF4F5F+F]qFhp,(F\qFb^nF+F+F]qFehz,$F\qFb^nQ1 ~2-gonal~numbersF$Q,EIS~A000326F$"$R$-Fc[l6#-F^cl6$,$*(FaepF4Fg\wF;FjdpF;F;Fbep 7#-F>6,Fav7%FB<%F]i`mFj[n/F_s-FL6%FbsFNFNFis77F+FinFinF+F+F+F5F5F4FdpFhpF2F:F]b lFddoFabnFc\nF`crFbbnF]`oFcirF_hioF^rF^rF^rFavQ,EIS~A008483F$"$Y%,$*&Ff]lFboF`h pF;F;7#-F>6,Fhcm7%FB<#/FB-FL6%Fd^lFd^lFeipFis77F+F5F4FdpFhpF2FiqF]blFjqFb\lFcuF d]mFc\nFcelF\`oFi`mFbbnF_blF[amF]hrFeclF`iio<(FdoF_`oFdhmF]amF``o,4FeqFboFfuFbo FjzFboF[rFboFjblFboFe\oFboF\qF+F]qFddoFc\lF+,*F\qF^fdl#FjdmFa^mF+F]qFe`bl-F^cl6 $,$*&,*FfclF4FdclFdpFa]oF4F\cuF+F+FhclF+Fi\[n/Fccl-F-6#,.F+F+F0F+FjemF+Fj]oF+F_ dlF+Fi]oF+F+FcfdlQYnumber~of~ways~to~make~n~cents~with~coins~of~1~1~6~centsF$Q, EIS~A008724F$Fdfw*$Ff]lFbo7#-F>6,Fav7%FB<#/FBFd[elFis77F+F5F4FdpFhpF2Fe]lFiqF:F ]blFgimFjqFddoFb\lFcdmFcuFabnFd]mF\_lFf`mFc\nFjjio<$Fdo,(FeqF+F;F+F]qF;FhuF]qFa vQ,EIS~A000027F$F_\r,**&FfglF4F*F+F+Fe[fmF^vFf[fmF4F*F;7#-F>6,F^hl7%FB<$F_eyF`[ gmFgn77FinFinFinFinFcelFg^mFad[l"%!y$"&[W""&k@&"'+:="'!y:'"(s?0#"(G!\n")+l(>#") +t+r"*'p4!G#"*s>XG("++^W6,Ffan7%FB<#/FB-Fe^l6#-FL6$FNFhedoFis77F+F+FdpFiqFg`mFdinFd\lF_ _mFj`go"%+?"%o\"&?B""&w0$"&ce("'C#)="'3qY"(_(e6"(s](G"(KO8(")/**p<")Gp"R%Ff]jo< &FdoF^uFa[qFe_fm-F^cl6$,$*&,(F\cuF+FfclF:FcclFhzF+FhclF+F[`fmF\`fm,$*(,&F;F+Fc` fmF+F+Fg`fmF;Fj`fmF+F;FavQ,EIS~A052528F$"$b%-F>6,F`el7%FBF\^joFgn77F+F+FiqFeclF ]`al"%SiFhicl"(gHj"")g&yD$"*++wD(",+%yy-="-+g(\x"\"/+;#H&fk9"0+[!)*)oNs%"2+)GGo Y!4k""3+S%4D"R%p5'"5+gx*HGYKWU#"7+?"H#3FlpiA5"8+g4W)RJ%z;5*[TK#eXo5Ff]jo<&FdoF^uF[ho,*FffnF+FhfnF+Fb[_lF+F[rF+*&F`_joF+F_glF+Fe_joF avQ,EIS~A052575F$Fe_[o,$*(F*F+,&FerF+F*FhpF+Ff]lFboF;7#-F>6,Fj]l7%FB<#/FB-FV6$F ^[[lFe_gmFis77FinFcuF\`oF`rFcirFihpFc_sF^`nFj`qF\dqFc_lFd^mFa]rFiinFcgzFb^wFdbl F]wFabjlF]`\mFf\`lFaajo<&F_uFcjn,(FeqF+F\cuF+F]qFd[w/F_pF\`o,&FhpF+F]qFgimFh[[l Q*~11~n~+~5F$Q,EIS~A017449F$"$%)*,$*&FcbcnF+FieoFboF;7#-F>6,Ffan7%FB<#/FB-FL6$F b]`nFdfoFis77F+FhpFb\lF_blFfinF]jnFbjv"%)3""%KC"%w`"&w<"F``[o"&'Hb"'%y=""'_RDF_ t"(!)o9""(K[U#"(3=6&")/zu5")%QWD#Fdbjo<&,(FeqFdpFfuF[yFjzF+FdoF^fhlF[gx,(*&F`ho F+F]qF+F+F`hoF+F]hoF+F^djoFavQ,EIS~A052951F$"%@5-Fc[l6#Fefm7#-F>6,Q:Stanley's~c hildren~roundsF$7%FB<#/FB-FF6#-FL6$FNF_gmFgn77F+FinF5F4Ff`mFb^mFfcoFh_`l"&WX$"' 3gJ"(Ss?$")+%*pN"*7.pK%"++7esc",W,;"**z".!30wO27"/![7E1B%>"0gD#4Oq6,Ffan7%FB<#/FB-Fe^l6#-FL6(Fd^lFd^lFNFNFNFNFis 77F+FinFinFinF+F5F4FdpF2F]blFabnFh`mFedmFa^mFcgqFccoF`d[oFbbgm"$*zFad[o"%!4#F]f jo<'FdoF[pF^pFfav,*FeqF+FjzF;F[rF5FjblF;-F^cl6$,$*&,*FcclFdqFdclF5FfclFdpF+F+F+ FhclF+#F;Fi`m/Fccl-F-6#,*F;F+F0F5FjemF;F_dlF+,$*(F`\_lF5,(Ff\_lF5Fa\_lFbo*$Fa\_ lFdpF[yF;Fg\_lF+F;FavQ,EIS~A024490F$"$'))*&FcioF+F*F+7#-F>6,F^hl7%FB<%F^[`lFad] nFghlFgn77FinF+F5F2Ff`mFj[qFdf`mF]fr"%;q"&gs$"'q9@"(DdF""(SG9)")hnxa"*=@-(Q"+I) *[jG",?nLF@#"-*\;C;y"".skln:\""/@qJf%fH""0S65W[l;"F_hjoF^rF^rF^rQ6rooted~set~pa rtitionsF$Q,EIS~A052889F$"$l)*&Ff]lF^vF_hpF;7$-F>6,Fhcm7%FB<#/FB-FL6&Fd^lFd^lFd ^lFbipFis77F+F4Fe]lFddoFg`mFj`mF`blFidmFbblF]jqF\_]lFabjlF[_m"$3$FaealFd]v"$D&F b]bo"$:(F_j^mF[jimFfijo<$Fdo,.FeqF2FfuF2F]qFb_cmFjflF+F`qF;F\qFdq,,F``uF`vF`qF^ fdlF\qF^aoF]q#FabnFjqF[chmF+,$F`qF^fdlQennumber~of~ways~to~make~n~cents~with~co ins~of~1~1~1~2~centsF$Q,EIS~A002623F$Fabjl-F>6,Fav7%FB<#/FB-FL6&FbipFd^lFd^lFd^ lFisF_jjoFfijoFcjjoFejjoFgjjoFavFijjo"$8%-Fc[l6#,&FjxFfrF[alF`[l7#-F>6,Fgjdm7%F B<#/FB-FF6#-FV6$F[`ioF\`ioFgn77F+FinF+FinF:FinFadqFin"%DEFin"&Xl(Fin"(X2W$Fin"* 043w"Fin",DuWM?"Fin"-D')p&*H#*Fin"/DGg&>0v)Fb[[p<',(Fb[wF+F\]lF+FjblF+FdoF[pFfa vFgblF^r,$*2F[vF\vF5FfrF`aioF+FdpFfrF]qF\vFeaioF+FhaioF+F[bioF+FbrQfnpermutatio ns~Sigma~without~fixed~point~such~that~Sigma^4=IdF$Q,EIS~A052503F$Fh`m-F_jl6#,$ *&Ff]lF+Fc[nF;F;7#-F>6,F^hl7%FB<%F]dhnFadwFcdwFgn77FinF5F2Fj`mFdco"%GH"&?&Q"'Si g")S#=6""*S'GHB"+!)G#*)[&"-+wt2P9".+3u8'QT"0+sI_6-I""1+oVBu?DW"3+ksU#\\>i""4+!G `nk/[pj"6+gH^*3aH3oE"8+g(\D_^tCZ(=""9+g&yPcaq.=ff&";+?$[uUSVtqdNy#F]][p<',*Fdez F+*&FgjflF+FfuF+F+*&F__imF+FjzF+F+F[rF+F_uFdfn/FioFj`mF]amF^r*&F]qF;Fe^nF+FavQ, EIS~A052878F$"$\),$*(,*F;F+F\\[lF+FjxF5F*F5F+F*F+,(F+F+F*F5FjxF+F;F\v7#-F>6,Ffv 7%FB<%F^sFddfnFivFgn77FinFinFinF2FeclFdgm"&!G<"'g&[%")S%*y8"*+Cq"\",+'p$H*>"-+# RR([!*"/+#R*\p[X"1+CAd*Rw]#"3+)oa"f^>/:"4++#zPN/s_(*"6+SiY$GvvI'z'"8+?B.R'ewx^l ]":+g`TQ+xK\J4-%"<+S!Rkz6W%=&\]'Q$">++su")>/#=g%*>:;I,$*(Fb_[pF;,*F+F+F*FboFjxF boF\\[lF;F+F*F+Ffr<*F_uF[pF^pFdz/Fi`oFi_[p,**&,*F`qFjflFcqF[yF\qF\_blF]qFjflF+F eqF+F+*&,*F\qF^fuFhzF+F]q!#eF`qF7F+FfuF+F+*&,(F]qF7F4F+F\qF^vF+FjzF+F+F`]coF+Fe gdl/FfpFdgmF^r,$*.F[vF\vFe^[lF+,&Fc^[lF+Fd^[lF+F;Fb^[lFfrF]qF]vFf^[lF+F`[lFavQ, EIS~A052744F$F[dim*&F_hpF;FeazF;7#-F>6,Fhcm7%FB<#/FB-FL6$FbipF^bzFis77F+FinF+Fi nF+F+F+F+F+F+F5F+F5F+F5F5F5F5F5F5F4Fja[p<(FdoF[pFfavFgblFhbl,2FeqF+FfuF5FjzF5F[ rF5FjblF5Fe\oF+F1F+F]qF;,*F]qF[^fl#Fe]lFf`mF+-F^cl6$,$*&,*FcclFdpFfclF+FdclF4F5 F+F+FhclF+Fb^flF^czF+F``uF`[l,$F]qF[^flQWnumber~of~ways~to~make~n~cents~with~co ins~of~2~5~centsF$Q,EIS~A008616F$F_^\m*(F*F5FcbcnF+FieoF;7#-F>6,F`el7%FB<#/FB-F L6%FNFNF\ccnFgn77FinFinF5Fd]mFhaqFgi_m"&?:"Fha`m"(![!e#")S'[k%"*+G(*G*",+;SP/#" -+%Qw\!\"/+%)fQHv7"0+_v!G#3d$"2+glA%oCr5"3+?*\_*)*)zU$"5+!G(\Qkl^l6"6+!3-fyJ'fe >%"8+S!zU'y+mEWf""9++;;rXJSmqxjF`c[p<(Fdfl/FbpFhaqF_^dlF_uFggoF[p,$Fb`emF`[l,$F `hoF`[lFavQ,EIS~A052638F$"$%e,$*&Ff]lF+,*F+F+F*F[yFjxF^vF9F4F;F;7#-F>6,Ffan7%FB <#/FB-Fe^l6#-FL6$F__pFg[glFis77F+F4FcdmF]^sFjn"%`8"%9h"&:w#"'VZ7"'vMc"(%GXD")Kt \6")bZ$>&"*k^fM#"+DHpf5"+FHv'y%","*\IA;#",q**Rqw*"-sgD)=T%".D#4t*G*>".1_!G9-!*F ^e[p<&FdoFd\x,*FeqF4FfuF^vFjzF[yF[rF+Fa`hm-F^cl6$,$*&,(Fd[wF+FcclF\bvFfclFcdmF+ FhclF+#F;Fbbl/Fccl-F-6#,*F+F+F0F[yFjemF^vFj]oF4,$*(,&F;F+-F-6$Fgg[p$"+gK!Q@#Fgd lF+F+,(*$F[h[pF5F2F[h[pFdp*$F[h[pF4FdqF;),(FdpF+F[h[pF4F`h[pF^vF]qF+F;FavQ,EIS~ A052981F$"%a5*(F_hpF+Ff]lF+Fb\`oF;7$-F>6,Fav7%FB<#/FB-Fe^l6#-FL6%FNFbipFd^lFis7 7F+F+F5FhpFgimF`crFb[sFd^mFjcl"$O'"%H9Feddm"%:s"&7i""&Gk$"&`=)"'AR="'pKT"'2'G*" (hl3#"(g%)o%Ffh[p<'FdoF^uFdflF^fioFggdm-F^cl6$,$*&,(FcclF+FfclF+FboF+F+FhclF+Fb ^`oFc^`o**,&F;F+F`_`oF+F+Fa_`oF+F__`oF;Fb_`oF+FavQ,EIS~A006054F$F`\fm-F>6,F`el7 %FB<#/FB-Fe^l6#F^[amFgn77F+F+FdpFi`mFfi[mFjhyFign"'S]j")g0T6"*!o"zI#"++_b&=&"-+ [%GQD";++Su&\eD\Pc19"Ffh[p<'FechoFdoF^u/FioFi`mFa[q*&F^j[ pF+F_glF+Fcj[pFavQ,EIS~A052658F$"$0'*&F*F+-Fc[l6#Fc^[nF+7#-F>6,F``x7%FBFdadoFgn 77FinF+F5F:FgclFbecl"%cq"&H**)"(KSG""),;N?"*g"[YN"+,AhOn"-G8Ls%Q""."3:Q5hI"/kO# *>8Rs"1&)\F**H.B="2Ofl4&o?p["4"[!)3H0#)[u8"5[S2ms2Ir(3%"7h:FU(*)f+1sF""8?Fu`xf1 pz?=%Fa\\pF^rF^rF^rQ;Labelled~trees~of~height~4F$Q,EIS~A052514F$Fihp*&F_hpF+Fh\ zF;7#-F>6,F`el7%FB<#/FB-Fe^l6#F_g_oFgn77F+F+F5FjqFa^mFhf[lFegm"&?b'"'?n%)")?zL7 F_ahm"++_f_N",+/Bw*o".+k%e*3X""/+Cy:i'G$"0++[k8o(z"2+gX>Oz]1#"3+?JpE#G.o&"5+?:, clLPa;"6+?fE$4[;)f3&"8++gp\J&3#eek"Fj]\p<&FdoF^uFdflFhagm*&-F^cl6$,$*&,&FboF+Fc clF+F+FhclF+Fh]nF[\_lF+F_glF+*(,&F;F+F`bgmF+F+F_bgmF;FcbemF+FavQ,EIS~A005443F$F ecrFf[jo7#-F>6,F^hl7%FB<$Ff\alF`[gmFgn77FinF+F2F\`oFc^mFeco"%e9Fajr"&'\"'R&\'"(kd7#"(L(3p")A0KA")NXur"*7DeH#"*dUzJ("+MH_CB"+"H*)4O(",S$H_CBFf[jo<%F ^uF_u,&FjfvF+F]cmF+F^rFa]joFavQ,EIS~A052758F$Fbjjo,&F^\ioF+FcioF;7#-F>6,Fav7%FB <%Fc`yF^[`lFghlFgn77FinF+F4F]blF[`nFf_l"$u'"%jK"&2q""&G[*"'&fi&"(F]`$")S3VB"*&) [Dj""+B#f?>""+-O=(4*",dws%Qs"-bj$>@*f".)*))Rl1:&"/:..;9*e%"0z8#zlXJUFaa\pF^rF^r F^rFavQ,EIS~A005493F$F^]]l,$*&,$Fee]nFboFfrF*F\vFfr7#-F>6,F^hl7%FBFfidnFgn77F+F +FhpFcirFjbgl"&TY"F[b]o")D1R6Fd``o",TIc$)p""-"eY+G%z"/\O@6lUT"1Dc,"z&=%Q#"3@"** pHNY4]""5*=-'eHrG1E5"6@.K'z?_$R%pv"8dMa?a'))\X*Q*f":D1*yL'\-\xU42&"<"3?C?`FL@?vGvcV"@h.0%[5'eJ!)o'phL%R%Fhb\pF^rF^r,$*,F[vF\vF5FfrFg[yFfrF]qF ]vFa^alF+F`[lFavQ,EIS~A052750F$Fbeio,,FfrF+F*FboFi\elF\vFjxFfr*&F*F+Fj\elFfrFfr 7#-F>6,Ffv7%FB<%Fg]_lF^aemFivFgn77FinFinF5FcelFh[m"%g")"'!))=#"(![;r"*S1Cs#",S1 ?&)>""-+c9")pf"/+g@4c>L"1+[;@0]Q?"3+OFtR=@q8"5+'HNlR@T1+""6+gXf-X*yI*)y"8+++3=6 XY1!zm":+!Gh0$))p'[1(H/'"<+g8>Qz3Rx6t">e">+gX$Gv7Am1EKPUf"@++sE@,jS#R!f9;TT',*F frF+F*F]vFi\elF\v*&,(FfrF+F*F\vFi\elF\vF+F*F+F;<'FdflF_uF[pFjgo,**&,*F\qF;F]qFd pFdpF+F`qF;F+FeqF+F+*&,(F]qFgimF1F+F\qFe]lF+FfuF+F+*&,&FgdlF+F]qFa`lF+FjzF+F+F[ rF+F^r,$**F[vF\v,&*$F^_elFfrF\v*&F`_elF+F^_elFfrFfrF+F]qF]vFa_elF+F\vFavQ,EIS~A 052736F$Fdd[l,$*&Ff]lF+Fj\elF;F;7#-F>6,Fav7%FB<#/FB-Fe^l6#-FV6$FNFainFis77F+Fhp FhjpF\j\l"$&)*"%Td"&hM$"'D]>"(*oO6"(4^i'")lRhQ"*"oe]A"+@"Q6,Fj]l7%FB<#/FB-FV6$F`^l-FL6$Fa`p Fd^lFis77FinFg`mFj`mF\`nF[\sFidmF^`nF_aqFfdqFa`nFb[rFi^rFd`nF\f[lF[[]lFfhvF^f[l F]`\mFhjhlFhe[lFeblFei\p<&F_u,(FeqF+FgdlF+F]qFhz/F\pFg`m/F_pFj`m,&F]blF+F]qFjqF e`lQ+~12~n~+~10F$Q,EIS~A017641F$"$***,$*&Ff]lF+,*F+F+F*F;F9FboF[alF+F;F;7#-F>6, Ffan7%FB<#/FB-Fe^l6#-FL6%FNFNFdghlFis77F+FinFinF5F+F+FhpFhpF2FcdmFf`mF\`oFfdmFe inFheqF]d^lFcc\mFd]enFdx"%x5FjjcmFij\p<'FdoF[pF^pFac[l,*FeqF+FfuFboF[rF;FjblF+- F^cl6$,$*&,*FddoF+FcclFf_vFfclF`rFdclFd]mF+FhclF+Fhbdm/Fccl-F-6#,*F+F+F0F;Fj]oF boF_dlF+,$*(,&F;F+-F-6$Fb\]p$"+w4X9kFgdlF+F+,(Ff\]pF+*$Ff\]pF4F2*$Ff\]pFdpF[yF; ),(F+F+*$Ff\]pF5F5F[]]pF;F]qF+F;FavQ,EIS~A052916F$"$**),$FdcvF;7#-F>6,Fj]l7%FB< #/FB-FV6$Fa`mFg_nFis77FinF4FhpFe]lF:FgimFddoFcdmFabnF\_lFc\nFcdoF`crF\`oFhjpF]_ lFbbnFddmF[`nF]`oFj[oFb]]p<&F_uFd\xFidx,(FeqF+F;F+F]qFbo,&F+F+F]qF5F_amQ)~2~n~+ ~1F$Q,EIS~A005408F$"$N**$Fa\wF57#-F>6,F^hl7%FB<%FcdwFg]_l/FbsFh`enFis77F+F5F4F2 F:Fb\lFg`mF[tF\`nF]^sFj`qFgaoFc[[lFhe[lFgax"$E%FejzF[]`lFgi_mFgafoF^gblFb^]pF^r F^rF^rFavQ,EIS~A022567F$"$_),$*&Ff]lF+,*F+F+F*FboFjxF;F[alF+F;F;7#-F>6,Ffan7%FB <#/FB-Fe^l6#-FL6$-FV6%FdipFNFd^lFNFis77F+F+F4Fe]lFcuF`rFh_qFd[[lF]gamFa[in"%3F" %gj"&P\""&"3N"&"R#)"'.N>"'gWX"(Ut1""(`n]#"(Xt)e")$)p#Q"F]_]p<'FdoF^uFdhm,*FeqF+ FjzF;F[rFboFjblF+Fj]`o-F^cl6$,$*&,*F^fzF+FcclF\bvFfclF]blFdclFiqF+FhclF+#F;Ffdq /Fccl-F-6#,*F+F+F0FboFjemF;F_dlF+,$*(,&F;F+-F-6$Fca]p$"+jm'yD%FgdlF+F+,(*$Fga]p F5F5Fga]pF5*$Fga]pFdpF[yF;),(F5F+Fga]pF+*$Fga]pF4F;F]qF+F;FavQ,EIS~A052967F$"%R 5,$*&FjfclF+,&F;F+F*FgimF;F;7#-F>6,Fav7%FB<#/FB-FL6$Fb]cl-Fe^l6#F`[[lFis77F+Fjq Fd\l"%_9"&sf""'#pv""(7E$>")K(e7#"*_g%QB"+slIsD",#Hs`HG"-7_4\7J".Kt/SPU$"/_1_S6m P"0srsXDF9%"1#*))*H+)*pb%"27y)H.yp7]"3Kf'GOenR^&"4___"*>Mk`1'"5sxx1>w2!>n'"6#\b X(4Q&34RtFcb]p<%FdoFddv,&FeqFgimFfuF;,$)FgimF^hoFjq,$)FgimF]q#FjqFgimFavQ,EIS~A 003954F$Fb_v,(-Feix6#,$-Fc[l6#,&F\vF+F*FfrF\vF;F\vF+F*Ffr7#-F>6,Fhio7%FBFigcnFg n77FinF+F+FdpF^blF_ejl"%_F"&3#R"'K+m")7*=G""*Cy8#G"+cy*)Rp"-%y#=m')=".W0-\th&"0 G(4Kq!z"="1si[Ugs`j"3GxDl4(R^Q#"43E,>ke?5d*"6'\5SWgc!z$)3%"8Cm!p)\O5aIA&="9;%)Q Z$>TI6Z4())F[e]pF^rF^r,$*.F[vF\vF5FfrFg[yFfr,$*&-Fc[l6#FieemF+,&FfrF+FjdnF;F+F\ vFfrF]qF]v),&F;F+FjdnF5FfjlF+FfrQHHierarchies,~Schroeder's~fourth~problemF$Q,EI S~A000311F$Fabl*(F*F+Ff]lF+FieoF;7#-F>6,F`el7%FB<#/FBF^f`mFgn77FinF+F5FjqFhaqFg i_mFjc[pFha`mF[d[pF\d[pF]d[pF^d[pF_d[pF`d[pFad[pFbd[pFcd[pFdd[pFed[pFfd[pFgd[pF ag]p<&F^uFdflF_uFggoFjd[pF[e[pFavQ,EIS~A014297F$"$1&F[f\p7#-F>6,Fav7%FB<%F^aemF d^gnFiefnFis77FinF+F5F2Fg`mFb^mF\\\n"%1="%e&)"&'eT"')41#"(=x.""(YMH&")QxHF"*Y(y ?9"*QqQX("+QIgPR",1n:F4#"-=g-==6"-ER&)=.g".urJCnB$F[f\p<&F^uFdflF_uFi[foF^rF\_e lFavQ,EIS~A006318F$Fcgz,$*&FjfclF+F[[mF;F;7#-F>6,Fav7%FB<#/FB-FL6$Fb]clFd[mFis7 7F+FhpFf`mFijpF\tFe_co"%?^"&![?"&?>)"'!oF$"(?2J""(!)GC&")?:(4#")!3')Q)"*?VaN$"+ !Gx@M""+?"4(o`",![O[Z@",?fM**e)"-!o$Q(fV$".?Z`*Qu8Fbi]p<%FdoF]`^nF[gx,$Fa^amFhp ,$F^]mFajlFavQ,EIS~A003947F$F^_m,(Fa]blF+Fb]blFboF9F+7#-F>6,F^hl7%FB<$/FB-FL6'F NFNFNF_sF_sF_eyFgn77FinFinFinFinFinFcenFhgn"%SHFg`v"&[7$Fdebm"'g9D"'+Kn"(_Pv""( k%oW")?m<6")+%=v#")g&Qo'"*OBUg""*wY,"Q"*+)=l*)F`[^p<)F_uF[pF^pFapFfavFcfy,(F_^x F+*&,&F\qF^vFjqF+F+FfuF+F+*&,(F\qF+F]qF^vF5F+F+FjzF+F+F^r,$*0F[vF\vF5FfrFcc_oF5 Fd_blF+F]qFehzFa^alF+FigyF+F`vFavQ,EIS~A052769F$F]hjm,$*&,0F;F+F*F+FjxF+F9F+F[a lF+F\[oF+Fj]pF+F;,.F+F+F*F+FjxF+F9F+F[alF+F\[oF+F+F;7#-F>6,QKWords~with~at~most ~5~consecutive~b-lettersF$7%FB<&FeeimFgeim/Fjeim-Fe^l6$Fheim1FSFhpF]fimFis77F+F 5FdpFiqFcuF[tFhdmF]jqF^\\mF\galFeafo"%O>Ff^]n"%.#*F_]^p<)FdoFa[qF]am/FfpF[t,0FeqF+FfuF+FjzF+F[rF+FjblF+Fe\oF+Fb`oF;Fd\elFhj _n-F^cl6$,$*&,."&n\'F+Fccl"&0k%Fdcl"&)))RFfcl"&/S%Fa]o"&KG$Fdao"&O2#F+FhclF+#F+ "'Pf?/Fccl-F-6#,0F;F+F0F+FjemF+Fj]oF+F_dlF+Fi]oF+F]boF+,$*(,.*$-F-6$Fj_^p$"+%e# QT]FgdlF2F1*$F_`^pF4F^v*$F_`^pF5Fbo*$F_`^pFdpF[y*$F_`^pFhpF\cuF_`^pF;F;,.F+F+F_ `^pF+Ff`^pF+Fd`^pF+Fc`^pF+Fe`^pF+F+)Fg`^pF]qF+F;QhnWords~with~at~most~5~consecu tive~b-letters,~Hexanacci~numbersF$Q,EIS~A001592F$Fddo,$*(F*F+Ff]lF;,&F`adlF+F; F+F+F;7#-F>6,F^hl7%FB<%Fj[nFjddnFc`yFgn77FinFinF5FcdmFgin"$D*"%c&)"&\#))"(cq+"" )"o7E""*SB4s""+TxOTD",%e!H&QS"-@m)37(o"/w'H?OhC""0&=6LpX*R#"1'ppR%>J))["3*[t[w% oQ]5"4cr:!*=UNRP#"5")*ec!R"*3))Gc"7!3VV))\@ljqR"F[a^p<&,**&,*FcqF;F`qF1F\qFd[wF ]qF1F+FeqF+F+*&,*Fd]mF+F`qF4F\qFd]mF]qFbbnF+FfuF+F+*&,(F\qF^vFhhdlF+F]qFebvF+Fj zF+F+F`]coF+FdflF_uF[pF^rF^ffmFavQ,EIS~A052874F$"$X),$*(F*F+,&F[bvF+F*F]blF+Ff] lFboF;7#-F>6,Fj]l7%FB<#/FB-FV6$F^[[lF^j\pFis77FinFc\nF[tF^_lF\\lFcbnFeinFbcoFab qF``nFcenFcblFi^rFf\vFadvFh_lFhdwFd`elF]jnF]gjlF`ijlF^c^p<&,(FeqF+FgdlF+F]qFd[w F_uFfehm/F_pF[t,&F]blF+F]qFgimFh[[lQ+~11~n~+~10F$Q,EIS~A017509F$"$*)**&FafmFdpF *F+7#-F>6,F^hl7%FB<$F^gmF]jxFgn77FinFinFinFinFinFcenFaiw"&!G9F[o"(/@Y"")g"[h""* ?:0">"++g$>U#",7(>0%G$"-oB+4^Z".gX6^HJ("0+O*f()>%>""1#>grCaK1#"2G0R9ves"5?NK0P#>?aX"Fad^p<)F_uF[pF^pFapFfavFcfy,,*&,4F\qF[bvF`qFe[qF[asF\fuF]q!# YFcelF+FcqF\cuF``sF+F]^tF2F^bvF+F+FeqF+F+*&,0F``sFa_\lF\qFi[vF]qFcelF`qFcirFcqF `crF[asFaqF]^tF[yF+FfuF+F+*&,.F]qFjqF``sF2FcqFc_lF`qFijpF\qF\_lF[asFeclF+FjzF+F +*&,,F`qFi[vF\qFja_nF[asF[yFcqFa_\lF]qFerF+F[rF+F+*&,*F]qF2FcqF+F`qF2F\qFgimF+F jblF+F+F^rF_cvFavQ,EIS~A052770F$Fbbdn-Fc[l6#,&Ff[fmF+F*F;7#-F>6,F``x7%FB<$/FBFh cio/F]cio-FL6$FN-FFF`]zFgn77F+FinF5F4FcuFcbnFgiclF_e\o"&C=""&X,)"'+ge"(,!)e%")C #R#Q"*,5hP$"+_tHWJ",XxQz2$"-c%>"*o:$".LG0fJQ$"/cMmO2zP"0$>C)Rg.R%"1!G#)o6'Q%H&F cf^pF^rF^rF^rQHMaps~of~f^2=f,Trees~of~height~at~most~1F$Q,EIS~A052506F$F]`o*&Fb ]glF+,(F;F+Fe]pF+F*F+F;7#-F>6,F`al7%FB<#/FB-Fe^l6#-FL6$FNF[^glFis77F+F+F+F+F+F+ F+F+F5F4FdpFhpF2Fe]lFiqF]blFddoFabnFg`mFh`mFddmF`h^p<+FdoF^u,(FeqF+Fb`oF+Fe`oF; FgblFi\lFhblFcbzFicelFjcel-F^cl6$,$*&,0Fccl!%wxFeao"&/N'Fdao"&)3uFa]o"&Ok)Fdcl" 'U35FfclF^hbn"&KW&F+F+FhclF+#F+"'*>q)/Fccl-F-6#,(F;F+F\boF+F0F+*(,&F;F+*$-F-6$F _j^p$"+TNWlzFgdlFe]lF+F+,&Fbj^pFa`lFcj^pF;F;),&F+F+*$Fcj^pF2F+F]qF+FavQ,EIS~A00 5709F$F`c`m,$*0,.F1F+F*FadqF9"%0AFjx!$+(F[al!%[KF\[oFiimF+F[[mF;Ff]lF;FieoF;F]^ pF;F_^pF;F`^pF;F;7#-F>6,Fg^p7%FB<#/FB-FVFadhmFis77F2Fc\nFa_lFiddl"%vA"&,A""&rr' "'hnP"(&fU@")hJJ7")^/Mr"*"o)*fT"+:dBQC",@&3,N9",JX"4u%)"-,io!z,&".Nw(e.yH"/"GQ! 3pq<"06M4hPW0""0@8.n#4(G'"1bV;r)GEv$F\[_p<)Ffehm/F_pFa_l/FioFiddl/FbpFj[_p/FfpF [\_p/FeoF2,0FeqFfd_lFfuF[fhmFjzF\fhmF[rF_h\lFjblF]fhmFe\oF+FcenF+,.F^]mF+F+F+F` hoF+FgcpF+FicpF+FjcpF+FjcpF^dpQ,EIS~A001553F$Ffaan*&Ff]lF+F_^pF;7$-F>6,Ffan7%FB <#/FB-Fe^l6#FainFis77F+FdpFf`mF_`nFbbbn"%+DFe[^n"&+D'"'+DJ"(+Dc""(+D"y")+D1R"*+ DJ&>"*+Dcw*"++D"G)[",+D19W#"-+DJq?7"-+Dc^.h".+D"yv^I"/+D1*ye_""/+DJXRHwFd]_p<%F doFhbwF^az,$F`jyFdp,$Ficp#FdpFhpFavQ,EIS~A005054F$Fagdn-F>6,F`el7%FB<#/FB-Fe^l6 #-FL6$Fd^lFf[mFgn77F+FdpF[amFhf[l"&+?""'++I"(+++*"*+++:$",++++E""-++++qc"/+++++ NG"1+++++Df:"2++++++bN*"4++++++v53'"6++++++]_nD%"8++++++]PkD>$":++++++++]^Sb#"< ++++++++]xV4<#">+++++++++vR\Q&>"@+++++++++Dw#p:c="B++++++++++Dw#p:c=Fd]_p<%FdoF hbwF\jy,$F_jyFdpF___pFavQ,EIS~A052675F$"$B',&F[ehoF+F`[`mF;7#-F>6,Fav7%FB<%Ff\a lF]fcmFghlFgn77FinF+FhpF\_lFcbnF[`l"$l'Fadhn"%0j"&r">"&D!e"'*4v""'Xt_"(Jhe""(&e mZ")RhJ9")&=")H%"*"44!H""*X$erQ"+zrth6"+Det&[$Faa_p<%F^uF_u,(FeqF2FfuF\cuFjzF+F ^rFjehoFavQ,EIS~A001047F$F`hdm*$Ff]lF[y7#-F>6,Fbe_l7%FB<$Fi[n/FB-FL6&F_sF_sF_sF _sFis77F+FdpF]blFf`mFddmFb[sF[emFcenF\gpF^jnFcc\mF\\vFi_joFj\`l"$!oF]euFgj[n"%S 6"%I8Fiafo"%r6,F^hl7%FB<%F^cxF[gmF_eyFgn77FinF5F4Fe]lF\`oFf_lFjg`o"%n$*"&( e%*"(r;4"")BX<9"*F^\/#"+Z^EXK",">NJ=c".jnp;P0""/()3%fo#G@"02d&>Qm0Y"26F'RO48j5" 3.=FeS`T2E"4Z1p8lKp5x'"6nA#olQY<.c=F^d_pF^rF^rFhdbmFavQ,EIS~A052877F$"$[)Fjfao7 #-F>6,Fav7%FB<%F_eglFca_lFf\alFgn77F+F+FdpFc\nFiin"%08"&1Q""',/<"(SOS#")xL9Q"*I Fbs'",TOYWI""-gQb.gF".D3*\_Ej"0QGcE56,Fdg\n7 %FB<'/%"dGFffp/FB-Fe^l6#-FV6&FfeimFheim%"cGF[h_p/Fah_pFffpFeeimFgeimFis77F+FdpF cuFgclF_e_lF]h\nFdjxF_h\nF`h\nF\i[lFah\n"(/V>%Fdiel")k)3r'F`f]nFfielFb^^oF]\goF hiel"-Wp!z([FFc^^oFcg_p<$FdoF]`^nF^]mF^]mQNWords~of~n~letters~in~a~four~letters ~alphabetF$Q,EIS~A000302F$Fiq-F>6,F`el7%FB<#/FBFd[mFgn77F+FdpF[tFh[mFdb]nF^h[l" (?"\H")g`d#)"+?:TUE",?Z"o7&*".+))es]!Q"0+s!R>Bu;"1+ca2LJO!)"3+7P#*>H))yT"5+sys: NW'o#3&[j16'"<+!3Q%fS`f#4xn*R%">+!3%H<&) eCP!f]vVL"@++k_$Q"3nzHs//]n#Fcg_p<$FdoFi\mF]]mF^]mFavQ,EIS~A047053F$F\gal,$*(F* F+FjfclF+F\iilF;F;7#-F>6,Fav7%FB<&Fa]clFcdw/FBFj_alF`\nFis77FinF+F4Fe]lFabnFj[o FfbqFiejlFe^dm"%$R""%jL"%>")"&,'>"&@t%"'VU6"'2eF"'dem"(@vg""(**3)Q"(>$p$*")P&>E #Faj_p<&F^uF_uFdhm,(FeqF+FfuF5FjzF;-F^cl6$F^cnFjjil,$**F`^dmF+F5FfrFa^dmF;Fc^dm F+F;FavQ,EIS~A001333F$F^_r*&F*F4FafmF47#-F>6,F^hl7%FB<$F^gmFh]blFgn77FinFinFinF inFinFinFhgnFdialF\bm"'+/o"(!o:q")!3/!y"*!3Ub$*",+#>T27"-/*fG7n"".!)oWO?Z#"/?$o 0CS*Q"0!)3`Lw3^'"2Cy(G4)**>:""3![#*=n"Q)3:#"4+[S:"G6,Q-Powers~of~12F$7%FB <$Ff\alF`aflFgn77F+FjqFe^mFigjmFd_^pFjf`l"(%)f)H")3=$e$"*'p")*H%"+_.yf^Fa\imF\d io".c#[/5;*)"0s!z`?$*p5"1k)[XY=RG""2ojeu:-2a""3;k.&*)eU)[="4#*pVSn56'=A"5/RC&)3 GLLiE"6[o#Hiq$***z%>$"7w@7vuW#**fP$QF[^`p<$Fdo,&FfuF+FeqFhzF^r,$*,F[vF\vF5FfrF] qF\v-Fc[l6#*&,&F+F+-F_jlFe`rF+F+F]qF+F+FigyF+FfrQ]oArrangement~of~n~distinguish able~objects~in~12distinguishable~urnsF$Q,EIS~A001021F$Fdco,$*&F*F4Ff]lF;F^v7#- F>6,F`el7%FB<#/FB-FL6&FNFNFd^lF__pFgn77FinFinFinFd]mFa^mFa_mFdelF]g[lFfablFcjdo FihemFhenFdjdoFejdoFfjdoFgjdoFhjdoFijdoFjjdoF[[eoF\[eoF[``p<'F_^dlF_uF[pF^pFahn F^[eoF4FavQ,EIS~A052619F$Fb[\o*$Ff]lF17$-F>6,F]c_l7%FB<$Fi[n/FB-FL6(F_sF_sF_sF_ sF_sF_sFis77F+F2Fc\nFb[sFd^mF[_mFdc_lFf[gnF^cgm"%-?Fgc_lFeafl"%)='"%o&)"&G;""&/ b""&\.#"&Mj#"&\O$"&/D%"&IJ&Fh``p<$Fdo,0FeqFcenF[asF;FcqFhhdlF`qFceflF\qFaeflF]q FbeflF^b]lF+,$*,FhuF+F\[lF+F]rF+F\clF+Fg\oF+Fjdjn,$F[asFjdjnQOArrangements~of~n ~unlabelled~objects~in~6~rowsF$Q,EIS~A000389F$Fjgal-F>6,Fbe_lF\a`pFaa`pFh``pF\b `pF^b`pF`b`pQMArrangements~of~n~labelled~objects~in~6~rowsF$Q,EIS~A001725F$Fc[z ,$Fc]mFabn7#-F>6,Q0Multiples~of~17F$7%FB<#/FB-FL6%-FV63FNFNFNFNFNFNFNFNFNFNFNFN FNFNFNFNFNFd^lFd^lFis77FinFabnFj`mFfdmFf^sFf^qFccqFahqFb`nFf\vFb^wFi_lFchdlFgfc lFhe[lFjgalFa[`lFfddlF^_mFj[[lFhivFgb`p<%F_uFj_gm,&FeqF+F]qF^fz,$F]qFabnFfc`pF[ c`pQ,EIS~A008599F$F`adn*&FdcmF+FcboF;7$-F>6,F`al7%FB<#/FB-Fe^l6#-FL6$FNF^dmFis7 7F+F+F+F+F5F4FdpF2F:FddoF\_lFh`mFj[oFihpFfinFdbnFdfwFi]clFc]]oFefw"$s)Fhc`p<'Fd oF^uFgblFi\lFe]]o-F^cl6$,$*&,(FcclFboFfclF:F2F+F+FhclF+FedoFfdo*(,&F;F+FaeoF+F+ F`eoF;FbeoF+FavF^^]oFhegl-F>6,F`el7%FBF]d`pFgn77F+F+F5F2FeclFa_mFc_flFb`sFgenFf acm")+s%*o"++/n<6",+c1R'>"-+![7it$".+ci*orw"0+?Z$***oo""1+?$))G2W&R"2+?f#e6,Fav7%FB<#/FB-FL6$Fb]clF]_p Fis77F+FdpFjqF_blFc^mF__mF`i^oFg^gn"%[()"&Wi#"&K(y"''>O#"')e3(Fe`\p"(#Hxj")w=8> ")GcRd"*%)o=s""*_1c;&"+c>o\:"+oe/\YF_f`p<%FdoFhbw,&FeqF4FfuF;,$)F4F^hoFdp,$Fgcp F`fhmFavQ,EIS~A003946F$F`d[oF`[`m7#-F>6,Q,Powers~of~2F$7%FB<$Ff\alFfh_lFgnF\h\n F`[`m<$Fdo,&FfuF+FeqFboF^rF\\`mQ\oArrangement~of~n~distinguishable~objects~in~2 distinguishable~urnsF$Feh\nF`j[m*(F*F+,*FjxFboF9F+F*F+F+F+F+Ff]lFbo7#-F>6,F`el7 %FB<#/FB-FL6$FN-FV6$FNFd[elFgn77FinF+F2Fd]mFhaqFhf[lFcjvFic\mFifoF]jhoFhelF[ean F[b`mF^jhoF_jhoF`jhoFc[hmFajhoFbjhoFcjho"5++!G`j,/e'[Feh`p<(F^uFid[pF_^dlF_uFdf nFjbmF_cmF]qFavQ,EIS~A052655F$Fhg^m-Fc[l6#,*F*F+FjxFfrF9F_emF[alF\e^l7#-F>6,Fhi bm7%FB<#/FB-FF6#-FV6&FdcemFecemFgcem-FFF]`ioFgn77F+F+F5FhpFcdmFfdmF\jnF_e_o"%&z $"&b(="&Y"**"'6nb"(<]I$")&Gb1#"*?(*RN""*hI(z#*"+@lbJm",(\^SH\"-]#emI!Q".&o]PXRI "/rALT07DFei`p<'FdoF^uFdflFggdm,,Fb[wF+*&,(Fe_mF+F\qF^vF]qFhhdlF+FfuF+F+*&,&Fe_ mF+F]qF1F+FjzF+F+FjblF2F[rF1F^r,$*6F[vF\vF5Ffr-Fc[l6##F[bvF[tF+FdpFfrF]qF\v-Fc[ l6#,$*&F2F`[lF]qF`[l#F]_lFgclF+-Fc[l6#,$*&F2FfrF]qFfr#FhpFcuF+-Fc[l6#,$*&F2F\ch mF]qF\chmF_emF+-Fc[l6#,$*&,&F;F+-F_jlFj`oF+F+F]qF+Fg[lF+F[bioF+FbrQeoSet~partit ions~with~all~blocks~of~size~<=4,~the~partition~function~G(n,4)~F$Q,EIS~A001681 F$F\_l,$*&Ff]lF+Ff_elF;F;7$-F>6,Ffan7%FB<#/FB-Fe^l6#-FL6$FN-FV6$FNFbf`mFis77F+F inF5F+FhpFhpFb\lF\_lFc\lF]^sFcblFgfcl"$<%"$G("%T8"%!Q#"%MV"%`x"&TS""&8_#"&Ub%Fb ]ap<&F_aelFdoFdflF[p-F^cl6$,$*(FcclF+,&F^vF+FcclF+F+FhclF+Fb^`oFdael,$*(,&F;F+F \belF+F+F`belF;FbbelF+F;FavQ,EIS~A052547F$"$$[-F>6,F`el7%FBFh]apFgn77F+FinFdpF2 FcenFhf[lFg`v"&gd*"(SMp"")!3]R#"*+GPv%"++Gh@))"-+smV(*>".+C9rK`%"0+#*\)31p6"1++ %e*\E7J"2+?fu8Pz1*"4+!)GG+jWwv#"5+![o7-#Hd*)*)"7+g@")yg;z.nI"9++))Q0QcKA*z5"Fb] ap<&FdoF[pFa[q,*F`[_lF+FhfnF+F\]lF+F[rF+*&F[_apF+F_glF+F`_apFavQ,EIS~A052672F$" $?'*&Ff]lF+,*F;F+F*F5FjxF;F\[oF+F;7#-F>6,Fav7%FB<#/FB-Fe^l6#-FV6$FNFcdflFis77F+ F+F+F+F+F5FdpFe]lFgimFcuFcdoFj`mFdinFd]qFd^mF_ealF^]pF]e[l"$&oF]a[n"%,;F\aap<(F doF^uFgblFi\lFhbl,*FeqF+F[rF;FjblF5Fe\oF;-F^cl6$,$*&,,FcclFdj`mFa]oF\\oFdclFf\` lFfclFc_vFd]qF+F+FhclF+#F+Fh]em/Fccl-F-6#,*F;F+F0F5FjemF;Fi]oF+*(,&F;F+-F-6$Ffb ap$"+U=XSlFgdlF+F+,(*$FibapF5F5FibapFbo*$FibapFhpF\cuF;),(F5F+FibapF;*$FibapFdp F+F]qF+FavQ,EIS~A005253F$Fegio*&F]^pF+F_^pF;7#-F>6,Ffan7%FB<#/FB-Fe^l6#-FL6$F]_ pFc`mFis77F+F5F]blF`blFebl"%]7F_`bn"&]7$"']i:"']7y"(]i!R")]7`>")]il(*"*]7G)["+] iSTC",]7.2A"",]i:N5'"-]7yv^I".]i!*ye_"".]7`%RHw"/]ilsp9QFdcap<%FdoF]amF^az,$F`j yF5,$FicpFddyFavQ,EIS~A020699F$FeduFjx7#-F>6,Fav7%FB<#/FBFdipFis77FinFinF+FinFi nFinFinFinFinFinFinFinFinFinFinFinFinFinFinFinFinFjx<&FeqF_uF[pFgblFinFinFavQ,E IS~A000038F$Fd\r,$*,Ff]lF;FeazF;F[`\lF;FbbclF;F_`\lF;F;7#-F>6,Fhcm7%FB<#/FB-FL6 'Fd^lF^bzF[a\lF\cclFba\lFis77F+F+F+F+F+F5F5F5F5F5FdpFdpFdpFdpFdpF2F2F2F2F2F:F\f apF]ds!&+A#Fads!&/X#Feds!&wL#FidsF_b^oF]es!& gb#Faes!&!3GFees!>#Fies!&g"HF]fs!&!oGFafs!&/*HFefs!&g(HFifs!&?&HF]gs!&w*HFags Ff[bpFegsFf[bpFigsFf[bpF]hsFf[bpFahs!&+]"FehsFg[bpFihs!&+i"F]is!&+9"Fais!&+Q"Fe is!&+E"FiisFb\bpF]jsF`\bpFajsF_\bpFejsF\\bpFijsF^\bpF][tF]\bpFa[tFi[bpFe[tF[\bp Fi[tFj[bpF]\tFh[bpFa\tFf\bpFe\tFc\bpFi\tFd\bpF]]tFa\bpFa]tFe\bpFe]tF_b^oFi]tFf[ bpF^^t!%+yFb^t!%+!*Ff^t!%CmFj^t!%'\&F]cp!%SWFjbp!%![$Fe`o!%SEFa_t!&+-"Fe_tFi\bp Fi_tFc]bpF]`tF[]bpFa`tFj\bpFe`tFg\bpFi`tFh\bpF]atF`a^oF`atFa]bpFcatFb]bpFfatF_] bpFiatF`]bpF\btFii]nF_btF`b]lFbbtFd[bpF[ctFbi]nF^ctFjflFactF_e`lFfjtF\]bpFijtF^ ]bpF\[uF]]bp,:-F^cl6$,$*(,TFdqF+FcclFdqFdclF`crFfclF`crFa]oF`crFdaoFe]lFdduF`cr FjduF`crFbduF`crFeaoFe]lF\euFe]lFhduFe]lFafuFj[oFbfuFj[oFcfuFj[oFdfuFcdoFefuFcd oFffuF`crFhfuF`crFifuFe]lFjfuF:F[guF:F`^vFe]lFgfuF`crFb^vF:F+F]buF+FhuF+F[f\n/F ccl-F-6#,.*$F0F`crF+Fd^^lF5FcfclF5Fe^^lF5Fi]oF5F+F+F+-F^cl6$,$*&,T!$l(F+Fccl"%X EFdclFcc^lFfcl"%&f#Fa]o"%&\#FdaoFe[`lFdduFf^bpFjduFcc^lFbduFg^bpFeaoFe^bpF\euFe [`lFhdu"%8QFafu"%JPFbfuFgc^lFcfu"%nNFdfu"%b>Fefu"%DBFffuFj[_pFhfu"%v@FifuF_\\lF jfuFgduF[guFfc^lF`^vFefwFgfuFic^lFb^vF^ifnF+FhclF+#F+Fgc\nFj]bpF+Fcq#F+"(++]"F` q#Fa_l"'++vF\q#"%*p#"'+]PF]q#"&td%F]`_p#"(hy)R"(+++&F+-F^cl6$,$*,,*F+F+FcclF5Fd clFdpFfclF4F+FccuF+FhuF+F\[lF+F]rF+#F;"(+](=F^czF+-F^cl6$,$**,*F\_blF+Fccl!#()F dclFedmFfcl!$H"F+F[huF+FhuF+F\[lF+#F+"'+]iF^czF+-F^cl6$,$*(,*!%RKF+Fccl!%1jFdcl "%:MFfcl"%5qF+F]buF+FhuF+#F+"(++D"F^czF+-F^cl6$,$*&,*!&T&HF+Fccl"'l(3"Fdcl"&!yK Ffcl"&v*oF+FhclF+FdabpF^czF+-F^cl6$,$*&,JFcclFa`lFdclFa`lFfclFa`lFdduF;FjduF;Fa ]oFa`lFdaoF[yFeaoF;FbfuF1F\euF[yFdfuF^vFafuF1FbduF;FcfuF1FhduF1FhfuF4FgfuF4Fffu F4FefuF4F^vF+F+FhclF+Fabbn/Fccl-F-6#,,Fd^^lF+FcfclF;Fe^^lF+Fi]oF;F+F+F+,$FcqF__ bpQjnnumber~of~ways~to~make~n~cents~with~coins~of~1~5~10~25~50~centsF$Q,EIS~A00 1300F$Fc[[lFa]bl7#-F>6,F^hl7%FB<$Ff[^pFf\alFgn77FinFinFinF2FeclFhyFgi_mF^bfn"&_ 2""&cA$"&g@*"'SMD"'Sen"(%=d<"(KGZ%")!3#=6")?^_F")?n%o'"*G@Vg""*/j-"Q"*![Kl*)Fa] bl<'F_uF[pF^p,&FhgoF+F\j\nF+FdzF^rF\]^pFavQ,EIS~A052771F$Fb^ap*&Ff]lF5,(F+F+F*F ^vF9F+F;7$-F>6,Ffan7%FB<#/FB-Fe^l6#-FL6%FNF`gzFd^lFis77F+F+FdpFgimF[tFe`qFc[z"$ j(Fia]o"%Ej"&:#="&[C&"'=5:"'R[V"(p?D""(*=0O")G2Q5"):,*)H")c^1')"*SZ"yC"*0Tb8(Fa dbp<&FdoF^uFa[q,(FeqF+FjzF^vF[rF+-F^cl6$,$*&,(F;F+FfclF5FcclFboF+FhclF+Feem/Fcc l-F-6#,(F+F+F0F^vFj]oF+*(,&F;F+-F-6$Fefbp$"+`N'HZ$FgdlF+F5,&FhfbpF4*$FhfbpF4F^v F;),&F4F+*$FhfbpF5F;F]qF+FavQ,EIS~A052545F$Fi\`l-F>6,F`el7%FB<#/FB-Fe^l6#-FL6%F NFd^lF`gzFgn77F+F+FiqF]^sFgg`l"&S5""'+3>"(?b%Q")SIe))"+!))ybH#",+?f)4m".+kKcN4# "/+)GO'yLs"1+7l(\^xq#"3+G\%)eB`"4""4+?b&4ZKT9Z"6+SYy%G3z$><#"8++/rO%)H"Q:j5"9+! oNTyPv"H@5b";++o$y&f)3I*[a9I"=++?2r.&=(*\@2gt"Fadbp<&FdoF^uF[ho,(F`[_lF+FbfhlF+ F[rF+*&F]fbpF+F_glF+FffbpFavQ,EIS~A052620F$"$m&,$*&Ff]lF+,*F+F+F*F;FjxF^vF9F+F; F;7#-F>6,Ffan7%FB<#/FB-Fe^l6#-FL6$-FV6$F_camFNFNFis77F+FinF4F5FgimFb\lFedmFeinF d`elF`c`m"$&*)"%Q="%VT"%i()"&`$>"&'\T"&$z!*"'Gf>"'6oU"'-Q#*"(2$3?Faibp<&FdoF[pF dhm,*FeqF+FfuF^vFjzF;F[rF+-F^cl6$,$*&,(F+F+FcclF\fuFfclF:F+FhclF+Fechl/Fccl-F-6 #,*F+F+F0F;FjemF^vFj]oF+,$*(,&F;F+-F-6$Ff[cp$"+s763YFgdlF+F+,(*$Fj[cpF5F2Fj[cpF +*$Fj[cpF4F^vF;),(F+F+Fj[cpF4F_\cpF;F]qF+F;FavQ,EIS~A052973F$F]]`l,$*&F*F+FieoF ;F^v7#-F>6,F`el7%FB<#/FB-FL6$FdfoF__pFgn77FinF4FjqFa^mF[h[oFegm"&?"p"'!on*")!)G [:"*S=py#"++o$Qd&Fg_em".+/$e)H%H"/+/fJw^w"1+7`%o$\U@"2+g$f`5[Fk"4+?&*\r$Rzc?"5+ !o$)4jQ*4$*p"7+![7ar!zd^6,Ffan7%FB<#/FB-Fe^l6#-FL6$ FcbwFNFis77F+F5FhpFb\lF`rF\dqFee[l"$w("%?@"%#z&"&Ce""&KK%"'7"=""')oA$"'+;))"(w& 3C"(_.e'")cy(z"");k6\"*W&)=M""*?*4mOF`^cp<&FdoF[ghnF]amFidx-F^cl6$Fc_imF_ghn*(, &F;F+FfghnF+F+FeghnF;FhghnF+FavQ,EIS~A052945F$"%/5,$*(F*F+,&FheclF+F*F+F+Ff]lFb oF;7#-F>6,Fj]l7%FB<#/FB-FV6$F`^lFg_nFis77FinFddoF`crF[`nFcirF[\oF]`nFf^qFjjpF`` nFjhqF_\rFc`nFhg\lF\j\lFdblFfd[mF\cyFjczF\ijlF]c`lFa`cp<&,(FeqF+F;F+F]qFhzF_u/F _pF`cr/F\pFddo,&F+F+F]qFjqFe`lQ*~12~n~+~1F$Q,EIS~A017533F$F^d`nFhdy7#-F>6,Fav7% FB<$Ffh_l/F_s-FL6$FNFg\alFgn77FinFinF5FjqFeclFjinFbjv"%W8F]tFe[alFhfo"&?j&"'o^8 "')[>$"'sau"(?.s"Fai[l"('*G"*)");S0?")Cm#[%")?Zh**Fhdy<&FdflF_uF[p,&FhgoF+Fjj_m F+F^rFi]cmFavQ,EIS~A001815F$Fb_lFic_o7#-F>6,Fav7%FB<$F_eyFfh_lFgn77FinFinF5F2Fb \lFi`mF\]sFd^mF^]\mF^\jmFi\\nFbjjm"%%4%"%!>)"&#Q;"&mF$"&Mb'"'q58"'U@E"''GC&"(u& [5Fic_o<&FdflF_uF[\`mF[pF^rF\\`mFavQ,EIS~A000918F$Fjdm,$FcadlFbo7#-F>6,Fgidm7%F B<#/FBF_camFis77FinF5F5F5F5F5F5F5F5F5F5F5F5F5F5F5F5F5F5F5F5F[dcp<%F_uF]am,&FeqF ;F5F+F5F5Q(All~2'sF$Q,EIS~A007395F$"%55,$F]``nF[y7#-F>6,Fj]l7%FB<#/FB-FV6$F`^lF ]^_oFis77FinFf`mF[tF]hrFb[sFf^sFijpFe`qF\dqFcgqFgaoF_^rF^]]lFadvFc[[lFccoFie[mF _gelFe_ilF_ejlF^\\mFhdcp<&F_u,(FeqF+F7F+F]qFhz/F\pFf`mF\d^p,&FiqF+F]qFjqFe`lQ*~ 12~n~+~8F$Q,EIS~A017617F$"$(**,$*$,(F;F+F*F+FjxF4F;F;7$-F>6,Fav7%FB<#/FB-Fe^l6# -FV6$FN-FL6$FNF__pFis77F+F+FdpFe]lF\_lF[amFjjpFjczFc[`l"%f6"%$o#"%gh"&4U""&*oK" &;`("'$Qt""'J$*R"'![>*"(tu6#"(8f([")K$G7"Fiecp<%FdoF^u,(FeqF4FfuF+FjzF;-F^cl6$, $*&,&F+F+FcclF2F+FhclF+F^h^l/Fccl-F-6#,(F;F+F0F+FjemF4,$*&,(F_emF+Ffh^lF\i^l*$, &F\i^lF+Ffh^lF_emF5F1F;)FdhcpFfjlF+F;FavQ,EIS~A006130F$F[ibo-F>6,F`el7%FBF`fcpF gn77F+F+FiqFc\lFc_[oFc_[l"&S)p"(!o$4"")gD[?"*?zd?%"++/2O(*"-+!)[()eC".+WtLh!o"0 +7$H3bN?"1+#>+=?fc'"3+S%p%\]GnA"4+!G\w3'=^N)"6++35dQwu/F$"8+SM&)*)ydM&ob8"9+ghj #H(pk#48$f";++[k&GSFr9V6,Fj]l7%FB<#/FB-FV6$Fc`imFg_nFis77FinFgimFc\nF]_lFj[oFfdmF[\oFc_sFd]qFa_lFjhmF ]fqFjhqFcblFd^rFf_lF\_]lFei\lFdblFaboFi\vFajcp<&F_uFf[[l,(FeqF+F;F+F]qFgdlFd__n ,&F+F+F]qF]blF[aimQ*~10~n~+~1F$Q,EIS~A017281F$F]erFf]io7#-F>6,F^hl7%FB<%FffalFa hlFb`yFgn77F+FinF5F4F[tFihy"%C:"&+E""'Gt9"(Obq"")?:oB"*S]gX$"+CE#\l&",SiwC')*". Wlb%fq="/![)[PSzP"0SE^)>it")"2gTK>p*Hu="3%=4$HVp[cX"5!))yfyIvx$o6"6!3=ry-()f0bJ Ff]ioF^rF^r*&F^[nFboF]qF;Q;Sequences~of~rooted~CyclesF$Q,EIS~A052830F$"$&z*.Ff] lF;FeazF;F[`\lF;F]`\lF;F_`\lF;Fa`\lF;7#-F>6,Fhcm7%FB<#/FB-FL6(Fd^lF^bzF[a\lF^a\ lFba\lFfa\lFis77F+F+F+F+F+F5F5F5F5F5FdpFdpFdpFdpFdpF2F2F2F2F2F]blF[]dp