 | ECS #207: Denumerant |
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1. Description
number of ways to make n cents with coins of 1 1 3 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
,Z \right) \right) \right) \right\}
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3. Coefficients
3.1. First terms
3.2. Recurrence
\displaystyle
\left\{ -2\,f \left( n \right) -2\,f \left( n+1 \right) -2\,f \left( n
+2 \right) +{n}^{2}+7\,n+12=0,f \left( 0 \right) =1,f \left( 1 \right)
=2 \right\}
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3.3. Closed form
\displaystyle
{\frac {8}{9}}+1/6\,{n}^{2}+5/6\,n+\sum _{\alpha={\rm RootOf} \left( {{
\rm \_Z}}^{2}+{\rm \_Z}+1 \right) }-1/9\,{\alpha}^{-n-1}
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3.4. Asymptotics
4. Ordinary generating function
\displaystyle
-{\frac {1}{ \left( -1+x \right) ^{2} \left( -1+{x}^{3} \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).