 | ECS #176: Denumerant
|
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).