 | ECS #177: Denumerant |
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1. Description
number of ways to make n cents with coins of 1 2 5 10 25 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 \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,Z,Z,Z,Z,Z \right) \right) \right) \right\}
other formats
3. Coefficients
3.1. First terms
3.2. Recurrence
\displaystyle
\left\{ 2686320+265518\,n-384\,f \left( n+3 \right) -2016\,f \left( n+
10 \right) -600\,f \left( n+34 \right) -840\,f \left( n+33 \right) -
1080\,f \left( n+32 \right) -1800\,f \left( n+29 \right) -1320\,f
\left( n+31 \right) -1560\,f \left( n+30 \right) -2184\,f \left( n+27
\right) -2016\,f \left( n+28 \right) +{n}^{4}-1320\,f \left( n+7
\right) -2304\,f \left( n+26 \right) +162\,{n}^{3}-2376\,f \left( n+25
\right) -24\,f \left( n+38 \right) -2400\,f \left( n+24 \right) -216\,
f \left( n+2 \right) -96\,f \left( n+1 \right) -600\,f \left( n+4
\right) -840\,f \left( n+5 \right) -1080\,f \left( n+6 \right) +9839\,
{n}^{2}-96\,f \left( n+37 \right) -2400\,f \left( n+20 \right) -2400\,f
\left( n+18 \right) -2400\,f \left( n+19 \right) -2400\,f \left( n+17
\right) -2400\,f \left( n+16 \right) -2400\,f \left( n+15 \right) -
2400\,f \left( n+14 \right) -2400\,f \left( n+22 \right) -2400\,f
\left( n+21 \right) -2400\,f \left( n+23 \right) -1560\,f \left( n+8
\right) -24\,f \left( n \right) -216\,f \left( n+36 \right) -384\,f
\left( n+35 \right) -2184\,f \left( n+11 \right) -2304\,f \left( n+12
\right) -1800\,f \left( n+9 \right) -2376\,f \left( n+13 \right) =0,f
\left( 0 \right) =1,f \left( 1 \right) =1,f \left( 2 \right) =2,f
\left( 3 \right) =2,f \left( 4 \right) =3,f \left( 5 \right) =4,f
\left( 6 \right) =5,f \left( 7 \right) =6,f \left( 8 \right) =7,f
\left( 9 \right) =8,f \left( 10 \right) =11,f \left( 11 \right) =12,f
\left( 12 \right) =15,f \left( 13 \right) =16,f \left( 14 \right) =19,
f \left( 15 \right) =22,f \left( 16 \right) =25,f \left( 17 \right) =28
,f \left( 18 \right) =31,f \left( 19 \right) =34,f \left( 20 \right) =
40,f \left( 21 \right) =43,f \left( 22 \right) =49,f \left( 23 \right)
=52,f \left( 24 \right) =58,f \left( 25 \right) =65,f \left( 26
\right) =71,f \left( 27 \right) =78,f \left( 28 \right) =84,f \left(
29 \right) =91,f \left( 30 \right) =102,f \left( 31 \right) =109,f
\left( 32 \right) =120,f \left( 33 \right) =127,f \left( 34 \right) =
138,f \left( 35 \right) =151,f \left( 36 \right) =162,f \left( 37
\right) =175 \right\}
other formats
3.3. Closed form
\displaystyle
{\frac {181317}{200000}}+\sum _{\alpha={\rm RootOf} \left( {{\rm \_Z}}^
{4}+{{\rm \_Z}}^{3}+{{\rm \_Z}}^{2}+{\rm \_Z}+1 \right) }{\frac {1}{
12500}}\, \left( 1-{\alpha}^{2}-\alpha+{\alpha}^{3} \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}{12500}}\, \left( 76-6\,{\alpha}^{2}+78\,\alpha+37
\,{\alpha}^{3} \right) {\alpha}^{-2-n} \left( n+1 \right) +\sum _{
\alpha={\rm RootOf} \left( {{\rm \_Z}}^{4}+{{\rm \_Z}}^{3}+{{\rm \_Z}}^
{2}+{\rm \_Z}+1 \right) }{\frac {1}{25000}}\, \left( -82+671\,{\alpha}^
{2}+1498\,\alpha+1252\,{\alpha}^{3} \right) {\alpha}^{-n-1}+\sum _{
\alpha={\rm RootOf} \left( {{\rm \_Z}}^{20}+{{\rm \_Z}}^{15}+{{\rm \_Z}
}^{10}+{{\rm \_Z}}^{5}+1 \right) }{\frac {1}{125}}\, \left( -{\alpha}^{
2}-2\,\alpha+2\,{\alpha}^{17}-{\alpha}^{8}+{\alpha}^{15}+3\,{\alpha}^{
19}+2\,{\alpha}^{14}-2\,{\alpha}^{6}-{\alpha}^{3}-{\alpha}^{7}+{\alpha}
^{13}+{\alpha}^{18}-{\alpha}^{11}-2-2\,{\alpha}^{10}-2\,{\alpha}^{5}
\right) {\alpha}^{-n-1}+\sum _{\alpha={\rm RootOf} \left( {{\rm \_Z}}^
{4}-{{\rm \_Z}}^{3}+{{\rm \_Z}}^{2}-{\rm \_Z}+1 \right) }{\frac {1}{200
}}\, \left( 2\,\alpha+4\,{\alpha}^{3}-3\,{\alpha}^{2}-6 \right) {\alpha
}^{-n-1}+{\frac {1}{160}}\, \left( -1 \right) ^{-n}n+{\frac {43}{320}}
\, \left( -1 \right) ^{-n}+{\frac {599}{15000}}\,{n}^{2}+{\frac {23521}
{60000}}\,n+{\frac {1}{60000}}\,{n}^{4}+{\frac {43}{30000}}\,{n}^{3}
other formats
3.4. Asymptotics
4. Ordinary generating function
\displaystyle
-{\frac {1}{ \left( -1+x \right) \left( -1+{x}^{2} \right) \left( -1+
{x}^{5} \right) \left( -1+{x}^{10} \right) \left( -1+{x}^{25}
\right) }}
other formats
5. References
6. Random structure
Search a combinatorial structure by: (firstTerms should be a sequence of integers, separated by commas).