 | ECS #199: Denumerant |
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1. Description
number of ways to make n cents with coins of 1 1 2 3 4 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
\right) \right) ,{\rm Sequence} \left( {\rm Prod} \left( Z,Z,Z,Z
\right) \right) \right) \right\}
other formats
3. Coefficients
3.1. First terms
3.2. Recurrence
\displaystyle
\left\{ -24\,f \left( n \right) -72\,f \left( n+1 \right) -120\,f
\left( n+2 \right) -144\,f \left( n+3 \right) -120\,f \left( n+4
\right) -72\,f \left( n+5 \right) -24\,f \left( n+6 \right) +{n}^{4}+
34\,{n}^{3}+431\,{n}^{2}+2414\,n+5040=0,f \left( 0 \right) =1,f \left(
1 \right) =2,f \left( 2 \right) =4,f \left( 3 \right) =7,f \left( 4
\right) =12,f \left( 5 \right) =18 \right\}
other formats
3.3. Closed form
\displaystyle
{\frac {2815}{3456}}+\sum _{\alpha={\rm RootOf} \left( {{\rm \_Z}}^{2}+
{\rm \_Z}+1 \right) }-1/27\,{\alpha}^{-1-n}+{\frac {1}{64}}\, \left( -1
\right) ^{-n}n+{\frac {11}{128}}\, \left( -1 \right) ^{-n}+{\frac {83}
{288}}\,{n}^{2}+{\frac {55}{64}}\,n+{\frac {1}{576}}\,{n}^{4}+{\frac {
11}{288}}\,{n}^{3}+\sum _{\alpha={\rm RootOf} \left( {{\rm \_Z}}^{2}+1
\right) }1/32\, \left( -1+\alpha \right) {\alpha}^{-1-n}
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3.4. Asymptotics
4. Ordinary generating function
\displaystyle
-{\frac {1}{ \left( -1+x \right) ^{2} \left( -1+{x}^{2} \right)
\left( -1+{x}^{3} \right) \left( -1+{x}^{4} \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).