 | ECS #189: Denumerant |
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1. Description
number of ways to make n cents with coins of 1 1 6 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,Z,Z,Z \right) \right) \right) \right\}
other formats
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) -2\,f \left( n+3 \right) -2\,f \left( n+4 \right) -2\,f
\left( n+5 \right) +{n}^{2}+13\,n+42=0,f \left( 0 \right) =1,f \left(
1 \right) =2,f \left( 2 \right) =3,f \left( 3 \right) =4,f \left( 4
\right) =5 \right\}
other formats
3.3. Closed form
\displaystyle
{\frac {77}{72}}+\sum _{\alpha={\rm RootOf} \left( {{\rm \_Z}}^{5}+{{
\rm \_Z}}^{4}+{{\rm \_Z}}^{3}+{{\rm \_Z}}^{2}+{\rm \_Z}+1 \right) }{
\frac {1}{72}}\, \left( 3\,{\alpha}^{4}+4\,{\alpha}^{3}+3\,{\alpha}^{2}
-5 \right) {\alpha}^{-1-n}+1/12\,{n}^{2}+2/3\,n
other formats
3.4. Asymptotics
4. Ordinary generating function
\displaystyle
-{\frac {1}{ \left( -1+x \right) ^{2} \left( -1+{x}^{6} \right) }}
other formats
5. References
6. Random structure
Search a combinatorial structure by: (firstTerms should be a sequence of integers, separated by commas).