 | ECS #233: Denumerant |
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1. Description
number of ways to make n cents with coins of 1 2 4 8 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) \right)
\right\}
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3. Coefficients
3.1. First terms
3.2. Recurrence
\displaystyle
\left\{ 6\,f \left( n \right) +18\,f \left( n+1 \right) +30\,f \left(
n+2 \right) +42\,f \left( n+3 \right) +48\,f \left( n+4 \right) +48\,f
\left( n+5 \right) +48\,f \left( n+6 \right) +48\,f \left( n+7
\right) +42\,f \left( n+8 \right) +30\,f \left( n+9 \right) +18\,f
\left( n+10 \right) +6\,f \left( n+11 \right) -{n}^{3}-39\,{n}^{2}-506
\,n-2184=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 \right\}
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3.3. Closed form
\displaystyle
{\frac {175}{256}}+\sum _{\alpha={\rm RootOf} \left( {{\rm \_Z}}^{2}+1
\right) }-{\frac {1}{128}}\, \left( \alpha+1 \right) {\alpha}^{-2-n}
\left( n+1 \right) +\sum _{\alpha={\rm RootOf} \left( {{\rm \_Z}}^{2}+
1 \right) }{\frac {1}{128}}\, \left( 7\,\alpha-6 \right) {\alpha}^{-1-n
}+{\frac {1}{256}}\, \left( -1 \right) ^{-n}{n}^{2}+{\frac {15}{256}}\,
\left( -1 \right) ^{-n}n+{\frac {49}{256}}\, \left( -1 \right) ^{-n}+
\sum _{\alpha={\rm RootOf} \left( {{\rm \_Z}}^{4}+1 \right) }1/32\,
\left( {\alpha}^{3}-1 \right) {\alpha}^{-1-n}+{\frac {15}{256}}\,{n}^{
2}+{\frac {295}{768}}\,n+{\frac {1}{384}}\,{n}^{3}
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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) }}
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5. References
6. Random structure
Search a combinatorial structure by: (firstTerms should be a sequence of integers, separated by commas).