 | ECS #243: Denumerant |
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1. Description
number of ways to make n cents with coins of 1 2 4 8 16 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 \right) \right) ,{\rm Sequence}
\left( {\rm Prod} \left( 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
\right) \right) \right) \right\}
other formats
3. Coefficients
3.1. First terms
3.2. Recurrence
\displaystyle
\left\{ 657720+92454\,n-576\,f \left( n+22 \right) -768\,f \left( n+21
\right) -96\,f \left( n+25 \right) -216\,f \left( n+24 \right) +{n}^{4
}-96\,f \left( n+1 \right) -384\,f \left( n+23 \right) -24\,f \left( n+
26 \right) -24\,f \left( n \right) +4871\,{n}^{2}+114\,{n}^{3}-216\,f
\left( n+2 \right) -384\,f \left( n+3 \right) -576\,f \left( n+4
\right) -768\,f \left( n+5 \right) -960\,f \left( n+6 \right) -1152\,f
\left( n+7 \right) -1320\,f \left( n+8 \right) -1440\,f \left( n+9
\right) -1512\,f \left( n+10 \right) -1536\,f \left( n+11 \right) -
1536\,f \left( n+12 \right) -1536\,f \left( n+13 \right) -1536\,f
\left( n+14 \right) -1536\,f \left( n+15 \right) -1512\,f \left( n+16
\right) -1440\,f \left( n+17 \right) -1320\,f \left( n+18 \right) -
1152\,f \left( n+19 \right) -960\,f \left( n+20 \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) =4,f \left( 5 \right) =4,f \left( 6
\right) =6,f \left( 7 \right) =6,f \left( 8 \right) =10,f \left( 9
\right) =10,f \left( 10 \right) =14,f \left( 11 \right) =14,f \left(
12 \right) =20,f \left( 13 \right) =20,f \left( 14 \right) =26,f
\left( 15 \right) =26,f \left( 16 \right) =36,f \left( 17 \right) =36,
f \left( 18 \right) =46,f \left( 19 \right) =46,f \left( 20 \right) =60
,f \left( 21 \right) =60,f \left( 22 \right) =74,f \left( 23 \right) =
74,f \left( 24 \right) =94,f \left( 25 \right) =94 \right\}
other formats
3.3. Closed form
\displaystyle
{\frac {4805}{12288}}\,n+{\frac {1271}{24576}}\,{n}^{2}+{\frac {31}{
12288}}\,{n}^{3}+{\frac {1}{24576}}\,{n}^{4}+{\frac {1}{12288}}\,
\left( -1 \right) ^{-n}{n}^{3}+{\frac {31}{8192}}\, \left( -1 \right)
^{-n}{n}^{2}+{\frac {635}{12288}}\, \left( -1 \right) ^{-n}n+\sum _{
\alpha={\rm RootOf} \left( {{\rm \_Z}}^{2}+1 \right) }-{\frac {1}{4096}
}\, \left( \alpha-1 \right) {\alpha}^{-3-n} \left( n+1 \right) \left(
n+2 \right) +\sum _{\alpha={\rm RootOf} \left( {{\rm \_Z}}^{2}+1
\right) }-{\frac {1}{4096}}\, \left( 27\,\alpha+29 \right) {\alpha}^{-
2-n} \left( n+1 \right) +\sum _{\alpha={\rm RootOf} \left( {{\rm \_Z}}^
{2}+1 \right) }{\frac {1}{4096}}\, \left( 195\,\alpha-166 \right) {
\alpha}^{-n-1}+\sum _{\alpha={\rm RootOf} \left( {{\rm \_Z}}^{4}+1
\right) }-{\frac {1}{512}}\, \left( \alpha+1 \right) {\alpha}^{-2-n}
\left( n+1 \right) +\sum _{\alpha={\rm RootOf} \left( {{\rm \_Z}}^{4}+
1 \right) }{\frac {1}{512}}\, \left( 15\,{\alpha}^{3}+2\,{\alpha}^{2}+2
\,\alpha-14 \right) {\alpha}^{-n-1}+\sum _{\alpha={\rm RootOf} \left( {
{\rm \_Z}}^{8}+1 \right) }{\frac {1}{64}}\, \left( -\alpha-{\alpha}^{2}
-{\alpha}^{3}-{\alpha}^{4}+{\alpha}^{7}-1 \right) {\alpha}^{-n-1}+{
\frac {3193}{16384}}\, \left( -1 \right) ^{-n}+{\frac {13175}{16384}}
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3.4. Asymptotics
4. Ordinary generating function
\displaystyle
-{\frac {1}{ \left( -1+x \right) \left( -1+{x}^{2} \right) \left( -1+
{x}^{4} \right) \left( -1+{x}^{8} \right) \left( -1+{x}^{16} \right)
}}
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5. References
6. Random structure
Search a combinatorial structure by: (firstTerms should be a sequence of integers, separated by commas).