 | ECS #198: Denumerant |
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1. Description
number of ways to make n cents with coins of 1 1 2 5 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( 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) \right) \right\}
other formats
3. Coefficients
3.1. First terms
3.2. Recurrence
\displaystyle
\left\{ 6\,f \left( n \right) +12\,f \left( n+1 \right) +12\,f \left(
n+2 \right) +12\,f \left( n+3 \right) +12\,f \left( n+4 \right) +6\,f
\left( n+5 \right) -{n}^{3}-21\,{n}^{2}-146\,n-336=0,f \left( 0
\right) =1,f \left( 1 \right) =2,f \left( 2 \right) =4,f \left( 3
\right) =6,f \left( 4 \right) =9 \right\}
other formats
3.3. Closed form
\displaystyle
{\frac {15}{16}}+\sum _{\alpha={\rm RootOf} \left( {{\rm \_Z}}^{4}+{{
\rm \_Z}}^{3}+{{\rm \_Z}}^{2}+{\rm \_Z}+1 \right) }1/25\, \left( {
\alpha}^{3}-1 \right) {\alpha}^{-1-n}+1/16\, \left( -1 \right) ^{-n}+{
\frac {9}{40}}\,{n}^{2}+{\frac {53}{60}}\,n+{\frac {1}{60}}\,{n}^{3}
other formats
3.4. Asymptotics
4. Ordinary generating function
\displaystyle
{\frac {1}{ \left( -1+x \right) ^{2} \left( -1+{x}^{2} \right) \left(
-1+{x}^{5} \right) }}
other formats
5. References
6. Random structure
Search a combinatorial structure by: (firstTerms should be a sequence of integers, separated by commas).