ECS #176: Denumerant
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1. Description

number of ways to make n cents with coins of 1 5 10 25 50 cents

2. Specification

This unlabelled structure is specified as S in

\displaystyle \left\{ S={\rm Prod} \left( {\rm Sequence} \left( Z \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,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\}
other formats

3. Coefficients

3.1. First terms

3.2. Recurrence

\displaystyle \left\{ 61324560-7800\,f \left( n+74 \right) -29760\,f \left( n+52 \right) -840\,f \left( n+4 \right) -29520\,f \left( n+53 \right) - 29160\,f \left( n+54 \right) -27360\,f \left( n+29 \right) -28680\,f \left( n+31 \right) -29160\,f \left( n+32 \right) -24\,f \left( n+86 \right) -1320\,f \left( n+81 \right) -840\,f \left( n+82 \right) - 28680\,f \left( n+55 \right) -21000\,f \left( n+23 \right) -22200\,f \left( n+24 \right) -23376\,f \left( n+25 \right) -6624\,f \left( n+75 \right) -5496\,f \left( n+76 \right) -4440\,f \left( n+77 \right) - 22200\,f \left( n+62 \right) -30000\,f \left( n+46 \right) -2640\,f \left( n+7 \right) -26520\,f \left( n+58 \right) -30000\,f \left( n+48 \right) -25560\,f \left( n+59 \right) -5496\,f \left( n+10 \right) - 29904\,f \left( n+51 \right) -1920\,f \left( n+6 \right) -21000\,f \left( n+63 \right) -6624\,f \left( n+11 \right) -30000\,f \left( n+49 \right) -23376\,f \left( n+61 \right) -30000\,f \left( n+44 \right) + 2772174\,n-10200\,f \left( n+14 \right) -11400\,f \left( n+15 \right) - 12600\,f \left( n+16 \right) -19800\,f \left( n+64 \right) -18600\,f \left( n+65 \right) -15000\,f \left( n+18 \right) -16200\,f \left( n+ 19 \right) -480\,f \left( n+83 \right) -240\,f \left( n+84 \right) -96 \,f \left( n+85 \right) -29520\,f \left( n+33 \right) -29760\,f \left( n+34 \right) -3480\,f \left( n+78 \right) -2640\,f \left( n+79 \right) -1920\,f \left( n+80 \right) -24504\,f \left( n+26 \right) -25560\,f \left( n+27 \right) -26520\,f \left( n+28 \right) -29904\,f \left( n+ 35 \right) -29976\,f \left( n+36 \right) -24\,f \left( n \right) +354\, {n}^{3}+{n}^{4}-96\,f \left( n+1 \right) -240\,f \left( n+2 \right) - 480\,f \left( n+3 \right) -17400\,f \left( n+20 \right) -7800\,f \left( n+12 \right) -9000\,f \left( n+13 \right) -30000\,f \left( n+37 \right) -30000\,f \left( n+38 \right) -30000\,f \left( n+39 \right) - 30000\,f \left( n+40 \right) -17400\,f \left( n+66 \right) -16200\,f \left( n+67 \right) -15000\,f \left( n+68 \right) -13800\,f \left( n+ 69 \right) -12600\,f \left( n+70 \right) -4440\,f \left( n+9 \right) - 28080\,f \left( n+30 \right) +46991\,{n}^{2}-30000\,f \left( n+45 \right) -13800\,f \left( n+17 \right) -29976\,f \left( n+50 \right) - 30000\,f \left( n+43 \right) -11400\,f \left( n+71 \right) -10200\,f \left( n+72 \right) -9000\,f \left( n+73 \right) -18600\,f \left( n+21 \right) -19800\,f \left( n+22 \right) -3480\,f \left( n+8 \right) - 1320\,f \left( n+5 \right) -24504\,f \left( n+60 \right) -30000\,f \left( n+42 \right) -30000\,f \left( n+41 \right) -30000\,f \left( n+ 47 \right) -27360\,f \left( n+57 \right) -28080\,f \left( n+56 \right) =0,f \left( 0 \right) =1,f \left( 1 \right) =1,f \left( 2 \right) =1,f \left( 3 \right) =1,f \left( 4 \right) =1,f \left( 5 \right) =2,f \left( 6 \right) =2,f \left( 7 \right) =2,f \left( 8 \right) =2,f \left( 9 \right) =2,f \left( 10 \right) =4,f \left( 11 \right) =4,f \left( 12 \right) =4,f \left( 13 \right) =4,f \left( 14 \right) =4,f \left( 15 \right) =6,f \left( 16 \right) =6,f \left( 17 \right) =6,f \left( 18 \right) =6,f \left( 19 \right) =6,f \left( 20 \right) =9,f \left( 21 \right) =9,f \left( 22 \right) =9,f \left( 23 \right) =9,f \left( 24 \right) =9,f \left( 25 \right) =13,f \left( 26 \right) =13,f \left( 27 \right) =13,f \left( 28 \right) =13,f \left( 29 \right) =13, f \left( 30 \right) =18,f \left( 31 \right) =18,f \left( 32 \right) =18 ,f \left( 33 \right) =18,f \left( 34 \right) =18,f \left( 35 \right) = 24,f \left( 36 \right) =24,f \left( 37 \right) =24,f \left( 38 \right) =24,f \left( 39 \right) =24,f \left( 40 \right) =31,f \left( 41 \right) =31,f \left( 42 \right) =31,f \left( 43 \right) =31,f \left( 44 \right) =31,f \left( 45 \right) =39,f \left( 46 \right) =39,f \left( 47 \right) =39,f \left( 48 \right) =39,f \left( 49 \right) =39, f \left( 50 \right) =50,f \left( 51 \right) =50,f \left( 52 \right) =50 ,f \left( 53 \right) =50,f \left( 54 \right) =50,f \left( 55 \right) = 62,f \left( 56 \right) =62,f \left( 57 \right) =62,f \left( 58 \right) =62,f \left( 59 \right) =62,f \left( 60 \right) =77,f \left( 61 \right) =77,f \left( 62 \right) =77,f \left( 63 \right) =77,f \left( 64 \right) =77,f \left( 65 \right) =93,f \left( 66 \right) =93,f \left( 67 \right) =93,f \left( 68 \right) =93,f \left( 69 \right) =93, f \left( 70 \right) =112,f \left( 71 \right) =112,f \left( 72 \right) = 112,f \left( 73 \right) =112,f \left( 74 \right) =112,f \left( 75 \right) =134,f \left( 76 \right) =134,f \left( 77 \right) =134,f \left( 78 \right) =134,f \left( 79 \right) =134,f \left( 80 \right) = 159,f \left( 81 \right) =159,f \left( 82 \right) =159,f \left( 83 \right) =159,f \left( 84 \right) =159,f \left( 85 \right) =187 \right\}
other formats

3.3. Closed form

\displaystyle {\frac {3987861}{5000000}}+\sum _{\alpha={\rm RootOf} \left( {{\rm \_Z} }^{20}-{{\rm \_Z}}^{15}+{{\rm \_Z}}^{10}-{{\rm \_Z}}^{5}+1 \right) }{ \frac {1}{500}}\, \left( -{\alpha}^{7}-4\,{\alpha}^{10}-4\,{\alpha}^{5} -7\,\alpha-{\alpha}^{6}-3\,{\alpha}^{15}-6\,{\alpha}^{11}-6\,{\alpha}^{ 13}-6\,{\alpha}^{12}-7\,{\alpha}^{2}-{\alpha}^{8}-6\,{\alpha}^{14}-7\,{ \alpha}^{3}-7\,{\alpha}^{4}-{\alpha}^{9}+3\,{\alpha}^{19}+3\,{\alpha}^{ 18}+3\,{\alpha}^{17}+3\,{\alpha}^{16}-3 \right) {\alpha}^{-1-n}+{\frac {2699}{375000}}\,{n}^{2}+{\frac {45773}{300000}}\,n+{\frac {1}{1500000} }\,{n}^{4}+{\frac {91}{750000}}\,{n}^{3}+\sum _{\alpha={\rm RootOf} \left( {{\rm \_Z}}^{25}+2\,{{\rm \_Z}}^{20}+2\,{{\rm \_Z}}^{15}+2\,{{ \rm \_Z}}^{10}+2\,{{\rm \_Z}}^{5}+1 \right) }{\frac {1}{100000}}\, \left( -9-9\,\alpha+25\,{\alpha}^{7}+7\,{\alpha}^{10}+7\,{\alpha}^{5}+ 7\,{\alpha}^{6}+23\,{\alpha}^{15}+7\,{\alpha}^{11}+41\,{\alpha}^{13}+41 \,{\alpha}^{12}+25\,{\alpha}^{2}+25\,{\alpha}^{8}+41\,{\alpha}^{14}+25 \,{\alpha}^{3}+25\,{\alpha}^{4}+25\,{\alpha}^{9}+25\,{\alpha}^{19}+25\, {\alpha}^{18}+25\,{\alpha}^{17}+23\,{\alpha}^{16}+7\,{\alpha}^{20}+9\,{ \alpha}^{24}+9\,{\alpha}^{23}+9\,{\alpha}^{22}+7\,{\alpha}^{21} \right) {\alpha}^{-n-2} \left( n+1 \right) +\sum _{\alpha={\rm RootOf} \left( {{\rm \_Z}}^{25}+2\,{{\rm \_Z}}^{20}+2\,{{\rm \_Z}}^{15}+2\,{{ \rm \_Z}}^{10}+2\,{{\rm \_Z}}^{5}+1 \right) }{\frac {1}{200000}}\, \left( -765+2645\,\alpha+2595\,{\alpha}^{7}+915\,{\alpha}^{10}+915\,{ \alpha}^{5}+2645\,{\alpha}^{6}+1955\,{\alpha}^{15}+3813\,{\alpha}^{11}+ 3649\,{\alpha}^{13}+3731\,{\alpha}^{12}+2595\,{\alpha}^{2}+2545\,{ \alpha}^{8}+3567\,{\alpha}^{14}+2545\,{\alpha}^{3}+2495\,{\alpha}^{4}+ 2495\,{\alpha}^{9}+2175\,{\alpha}^{19}+2225\,{\alpha}^{18}+2275\,{ \alpha}^{17}+2325\,{\alpha}^{16}+595\,{\alpha}^{20}+783\,{\alpha}^{24}+ 801\,{\alpha}^{23}+819\,{\alpha}^{22}+837\,{\alpha}^{21} \right) { \alpha}^{-1-n}+\sum _{\alpha={\rm RootOf} \left( {{\rm \_Z}}^{4}+{{\rm \_Z}}^{3}+{{\rm \_Z}}^{2}+{\rm \_Z}+1 \right) }-{\frac {1}{1875000}}\, \left( 1+2\,\alpha+3\,{\alpha}^{2}+4\,{\alpha}^{3} \right) {\alpha}^{- 4-n} \left( n+1 \right) \left( n+2 \right) \left( n+3 \right) +\sum _ {\alpha={\rm RootOf} \left( {{\rm \_Z}}^{4}+{{\rm \_Z}}^{3}+{{\rm \_Z}} ^{2}+{\rm \_Z}+1 \right) }{\frac {1}{625000}}\, \left( -44-87\,\alpha- 129\,{\alpha}^{2}+45\,{\alpha}^{3} \right) {\alpha}^{-n-3} \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}{1250000}}\, \left( -3239-6306\,\alpha+7010\,{\alpha}^{2}+3415\,{ \alpha}^{3} \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}{1250000}}\, \left( -29541+108765\,\alpha +68975\,{\alpha}^{2}+32780\,{\alpha}^{3} \right) {\alpha}^{-1-n}
other formats

3.4. Asymptotics

4. Ordinary generating function

\displaystyle -{\frac {1}{ \left( -1+x \right) \left( -1+{x}^{5} \right) \left( -1+ {x}^{10} \right) \left( -1+{x}^{25} \right) \left( -1+{x}^{50} \right) }}
other formats

5. References

EIS A001300

6. Random structure


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

Generated: 2010-02-09 22:52:11 in 19. seconds of elapsed time.
Based on commit 9d1e479..., Fri Jul 10 14:52:30 2009 +0200.
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