ECS #184: Denumerant
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1. Description

number of ways to make n cents with coins of 2 5 10 20 50 cents

2. Specification

This unlabelled structure is specified as S in

\displaystyle \left\{ S={\rm Prod} \left( {\rm Sequence} \left( {\rm Prod} \left( Z, Z \right) \right) ,{\rm Sequence} \left( {\rm Prod} \left( Z,Z,Z,Z,Z \right) \right) ,{\rm Sequence} \left( {\rm Prod} \left( Z,Z,Z,Z,Z,Z, Z,Z,Z,Z \right) \right) ,{\rm Sequence} \left( {\rm Prod} \left( Z,Z,Z ,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z \right) \right) ,{\rm Sequence} \left( {\rm Prod} \left( Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z ,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z \right) \right) \right) \right\}
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3. Coefficients

3.1. First terms

3.2. Recurrence

\displaystyle \left\{ 50965320-48000\,f \left( n+49 \right) +2412982\,n-45840\,f \left( n+55 \right) -44760\,f \left( n+56 \right) -47976\,f \left( n+ 50 \right) -33576\,f \left( n+20 \right) -41880\,f \left( n+24 \right) -40080\,f \left( n+23 \right) -44760\,f \left( n+26 \right) -43440\,f \left( n+25 \right) -45840\,f \left( n+27 \right) -47976\,f \left( n+ 32 \right) -48000\,f \left( n+33 \right) -48000\,f \left( n+34 \right) -16800\,f \left( n+69 \right) -14424\,f \left( n+70 \right) -12120\,f \left( n+71 \right) -9936\,f \left( n+72 \right) -7920\,f \left( n+73 \right) -6120\,f \left( n+74 \right) -4560\,f \left( n+75 \right) - 3240\,f \left( n+76 \right) -2160\,f \left( n+77 \right) -1320\,f \left( n+78 \right) -720\,f \left( n+79 \right) -336\,f \left( n+80 \right) -120\,f \left( n+81 \right) -24\,f \left( n+82 \right) -47664 \,f \left( n+52 \right) -47280\,f \left( n+53 \right) -46680\,f \left( n+54 \right) -4560\,f \left( n+7 \right) -48000\,f \left( n+38 \right) -48000\,f \left( n+37 \right) -48000\,f \left( n+47 \right) -48000\,f \left( n+46 \right) -48000\,f \left( n+48 \right) -48000\,f \left( n+ 42 \right) -48000\,f \left( n+43 \right) -48000\,f \left( n+44 \right) -48000\,f \left( n+45 \right) -28800\,f \left( n+18 \right) -24000\,f \left( n+16 \right) -26400\,f \left( n+17 \right) -35880\,f \left( n+ 21 \right) -38064\,f \left( n+22 \right) -31200\,f \left( n+19 \right) -47880\,f \left( n+51 \right) -40080\,f \left( n+59 \right) -33576\,f \left( n+62 \right) -38064\,f \left( n+60 \right) -35880\,f \left( n+ 61 \right) -47664\,f \left( n+30 \right) -47880\,f \left( n+31 \right) -47280\,f \left( n+29 \right) -46680\,f \left( n+28 \right) +42839\,{n} ^{2}-24\,f \left( n \right) +338\,{n}^{3}+{n}^{4}-31200\,f \left( n+63 \right) -28800\,f \left( n+64 \right) -26400\,f \left( n+65 \right) - 24000\,f \left( n+66 \right) -21600\,f \left( n+67 \right) -19200\,f \left( n+68 \right) -43440\,f \left( n+57 \right) -41880\,f \left( n+ 58 \right) -14424\,f \left( n+12 \right) -16800\,f \left( n+13 \right) -120\,f \left( n+1 \right) -6120\,f \left( n+8 \right) -12120\,f \left( n+11 \right) -21600\,f \left( n+15 \right) -7920\,f \left( n+9 \right) -19200\,f \left( n+14 \right) -3240\,f \left( n+6 \right) - 9936\,f \left( n+10 \right) -720\,f \left( n+3 \right) -48000\,f \left( n+36 \right) -336\,f \left( n+2 \right) -48000\,f \left( n+40 \right) -1320\,f \left( n+4 \right) -2160\,f \left( n+5 \right) -48000 \,f \left( n+35 \right) -48000\,f \left( n+39 \right) -48000\,f \left( n+41 \right) =0,f \left( 0 \right) =1,f \left( 1 \right) =0,f \left( 2 \right) =1,f \left( 3 \right) =0,f \left( 4 \right) =1,f \left( 5 \right) =1,f \left( 6 \right) =1,f \left( 7 \right) =1,f \left( 8 \right) =1,f \left( 9 \right) =1,f \left( 10 \right) =3,f \left( 11 \right) =1,f \left( 12 \right) =3,f \left( 13 \right) =1,f \left( 14 \right) =3,f \left( 15 \right) =3,f \left( 16 \right) =3,f \left( 17 \right) =3,f \left( 18 \right) =3,f \left( 19 \right) =3,f \left( 20 \right) =7,f \left( 21 \right) =3,f \left( 22 \right) =7,f \left( 23 \right) =3,f \left( 24 \right) =7,f \left( 25 \right) =7,f \left( 26 \right) =7,f \left( 27 \right) =7,f \left( 28 \right) =7,f \left( 29 \right) =7,f \left( 30 \right) =13,f \left( 31 \right) =7,f \left( 32 \right) =13,f \left( 33 \right) =7,f \left( 34 \right) =13,f \left( 35 \right) =13,f \left( 36 \right) =13,f \left( 37 \right) =13,f \left( 38 \right) =13,f \left( 39 \right) =13,f \left( 40 \right) =22,f \left( 41 \right) =13,f \left( 42 \right) =22,f \left( 43 \right) =13, f \left( 44 \right) =22,f \left( 45 \right) =22,f \left( 46 \right) =22 ,f \left( 47 \right) =22,f \left( 48 \right) =22,f \left( 49 \right) = 22,f \left( 50 \right) =35,f \left( 51 \right) =22,f \left( 52 \right) =35,f \left( 53 \right) =22,f \left( 54 \right) =35,f \left( 55 \right) =35,f \left( 56 \right) =35,f \left( 57 \right) =35,f \left( 58 \right) =35,f \left( 59 \right) =35,f \left( 60 \right) =53,f \left( 61 \right) =35,f \left( 62 \right) =53,f \left( 63 \right) =35, f \left( 64 \right) =53,f \left( 65 \right) =53,f \left( 66 \right) =53 ,f \left( 67 \right) =53,f \left( 68 \right) =53,f \left( 69 \right) = 53,f \left( 70 \right) =77,f \left( 71 \right) =53,f \left( 72 \right) =77,f \left( 73 \right) =53,f \left( 74 \right) =77,f \left( 75 \right) =77,f \left( 76 \right) =77,f \left( 77 \right) =77,f \left( 78 \right) =77,f \left( 79 \right) =77,f \left( 80 \right) =108,f \left( 81 \right) =77 \right\}
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3.3. Closed form

\displaystyle {\frac {3198151}{8000000}}+\sum _{\alpha={\rm RootOf} \left( {{\rm \_Z} }^{40}+{{\rm \_Z}}^{30}+{{\rm \_Z}}^{20}+{{\rm \_Z}}^{10}+1 \right) }{ \frac {1}{250}}\, \left( -1-{\alpha}^{4}-{\alpha}^{24}-{\alpha}^{14}+{ \alpha}^{26}+{\alpha}^{36}+{\alpha}^{21}+{\alpha}^{31}-{\alpha}^{2}+{ \alpha}^{23}-{\alpha}^{10}+{\alpha}^{29}+{\alpha}^{39}+{\alpha}^{33}-{ \alpha}^{20}+{\alpha}^{28}+{\alpha}^{25}+{\alpha}^{35}-{\alpha}^{12}-{ \alpha}^{22}+{\alpha}^{27}+{\alpha}^{37}+{\alpha}^{38} \right) {\alpha} ^{-1-n}+\sum _{\alpha={\rm RootOf} \left( {{\rm \_Z}}^{5}+2\,{{\rm \_Z} }^{4}+2\,{{\rm \_Z}}^{3}+2\,{{\rm \_Z}}^{2}+2\,{\rm \_Z}+1 \right) }{ \frac {1}{6000000}}\, \left( 41\,{\alpha}^{4}+29+25\,{\alpha}^{2}+33\,{ \alpha}^{3}+37\,\alpha \right) {\alpha}^{-4-n} \left( n+1 \right) \left( n+2 \right) \left( n+3 \right) +\sum _{\alpha={\rm RootOf} \left( {{\rm \_Z}}^{5}+2\,{{\rm \_Z}}^{4}+2\,{{\rm \_Z}}^{3}+2\,{{\rm \_Z}}^{2}+2\,{\rm \_Z}+1 \right) }-{\frac {1}{4000000}}\, \left( 2351\, {\alpha}^{4}+1687+2027\,{\alpha}^{2}+1331\,{\alpha}^{3}+2659\,\alpha \right) {\alpha}^{-3-n} \left( n+1 \right) \left( n+2 \right) +\sum _ {\alpha={\rm RootOf} \left( {{\rm \_Z}}^{5}+2\,{{\rm \_Z}}^{4}+2\,{{ \rm \_Z}}^{3}+2\,{{\rm \_Z}}^{2}+2\,{\rm \_Z}+1 \right) }{\frac {1}{ 4000000}}\, \left( 58631\,{\alpha}^{4}+24463+72751\,{\alpha}^{2}+34551 \,{\alpha}^{3}+45919\,\alpha \right) {\alpha}^{-n-2} \left( n+1 \right) +\sum _{\alpha={\rm RootOf} \left( {{\rm \_Z}}^{5}+2\,{{\rm \_Z}}^{4}+2\,{{\rm \_Z}}^{3}+2\,{{\rm \_Z}}^{2}+2\,{\rm \_Z}+1 \right) }-{\frac {1}{8000000}}\, \left( 430871\,{\alpha}^{4}+58055+266551\,{ \alpha}^{2}-199929\,{\alpha}^{3}-512969\,\alpha \right) {\alpha}^{-1-n} +\sum _{\alpha={\rm RootOf} \left( {{\rm \_Z}}^{10}+1 \right) }{\frac { 1}{320}}\, \left( \alpha-{\alpha}^{2}+{\alpha}^{3}-{\alpha}^{4}+{\alpha }^{5}+{\alpha}^{6}+{\alpha}^{7}+{\alpha}^{8}+{\alpha}^{9}-1 \right) { \alpha}^{-1-n}+\sum _{\alpha={\rm RootOf} \left( {{\rm \_Z}}^{4}-{{\rm \_Z}}^{3}+{{\rm \_Z}}^{2}-{\rm \_Z}+1 \right) }{\frac {1}{200000}}\, \left( -2-{\alpha}^{2}+3\,{\alpha}^{3}-\alpha \right) {\alpha}^{-3-n} \left( n+1 \right) \left( n+2 \right) +\sum _{\alpha={\rm RootOf} \left( {{\rm \_Z}}^{4}-{{\rm \_Z}}^{3}+{{\rm \_Z}}^{2}-{\rm \_Z}+1 \right) }{\frac {1}{100000}}\, \left( -123+45\,{\alpha}^{2}+84\,{ \alpha}^{3}+43\,\alpha \right) {\alpha}^{-n-2} \left( n+1 \right) + \sum _{\alpha={\rm RootOf} \left( {{\rm \_Z}}^{4}-{{\rm \_Z}}^{3}+{{ \rm \_Z}}^{2}-{\rm \_Z}+1 \right) }{\frac {1}{100000}}\, \left( -1377- 648\,{\alpha}^{2}+2191\,{\alpha}^{3}+3094\,\alpha \right) {\alpha}^{-1- n}+{\frac {9839}{2400000}}\,{n}^{2}+{\frac {6583}{80000}}\,n+{\frac {1} {2400000}}\,{n}^{4}+{\frac {29}{400000}}\,{n}^{3}
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3.4. Asymptotics

4. Ordinary generating function

\displaystyle -{\frac {1}{ \left( -1+{x}^{2} \right) \left( -1+{x}^{5} \right) \left( -1+{x}^{10} \right) \left( -1+{x}^{20} \right) \left( -1+{x}^ {50} \right) }}
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5. References

EIS A001319

6. Random structure


Search a combinatorial structure by: (firstTerms should be a sequence of integers, separated by commas).

Generated: 2009-11-21 09:20:36 in 8. seconds of elapsed time.
Based on commit 9d1e479..., Fri Jul 10 14:52:30 2009 +0200.
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