 | ECS #214: Denumerant |
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1. Description
number of ways to make n cents with coins of 2 7 cents
2. Specification
This unlabelled structure is specified as S in
\displaystyle
\left\{ S={\rm Prod} \left( {\rm Sequence} \left( {\rm Prod} \left( Z,
Z \right) \right) ,{\rm Sequence} \left( {\rm Prod} \left( Z,Z,Z,Z,Z,Z
,Z \right) \right) \right) \right\}
other formats
3. Coefficients
3.1. First terms
3.2. Recurrence
\displaystyle
\left\{ 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) +2\,f \left( n+6 \right) +f \left( n+7 \right) -n-8
=0,f \left( 0 \right) =1,f \left( 1 \right) =0,f \left( 2 \right) =1,f
\left( 3 \right) =0,f \left( 4 \right) =1,f \left( 5 \right) =0,f
\left( 6 \right) =1 \right\}
other formats
3.3. Closed form
\displaystyle
{\frac {9}{28}}+1/14\,n+\sum _{\alpha={\rm RootOf} \left( {{\rm \_Z}}^{
6}+{{\rm \_Z}}^{5}+{{\rm \_Z}}^{4}+{{\rm \_Z}}^{3}+{{\rm \_Z}}^{2}+{
\rm \_Z}+1 \right) }1/49\, \left( 4\,{\alpha}^{5}+{\alpha}^{4}+5\,{
\alpha}^{3}+2\,{\alpha}^{2}+6\,\alpha+3 \right) {\alpha}^{-1-n}+1/4\,
\left( -1 \right) ^{-n}
other formats
3.4. Asymptotics
4. Ordinary generating function
\displaystyle
{\frac {1}{ \left( -1+{x}^{2} \right) \left( -1+{x}^{7} \right) }}
other formats
5. References
6. Random structure
Search a combinatorial structure by: (firstTerms should be a sequence of integers, separated by commas).