 | ECS #205: Denumerant |
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1. Description
number of ways to make n cents with coins of 1 1 1 2 2 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( Z \right) ,{\rm
Sequence} \left( {\rm Prod} \left( Z,Z \right) \right) ,{\rm Sequence}
\left( {\rm Prod} \left( Z,Z \right) \right) ,{\rm Sequence} \left( {
\rm Prod} \left( Z,Z,Z \right) \right) \right) \right\}
other formats
3. Coefficients
3.1. First terms
3.2. Recurrence
\displaystyle
\left\{ 120\,f \left( n \right) +360\,f \left( n+1 \right) +480\,f
\left( n+2 \right) +360\,f \left( n+3 \right) +120\,f \left( n+4
\right) -15120-3325\,{n}^{2}-485\,{n}^{3}-35\,{n}^{4}-{n}^{5}-11274\,n
=0,f \left( 0 \right) =1,f \left( 1 \right) =3,f \left( 2 \right) =8,f
\left( 3 \right) =17 \right\}
other formats
3.3. Closed form
\displaystyle
{\frac {1529}{1728}}+{\frac {5}{64}}\, \left( -1 \right) ^{-n}+{\frac {
1}{64}}\, \left( -1 \right) ^{-n}n+{\frac {25}{36}}\,{n}^{2}+{\frac {
427}{320}}\,n+{\frac {5}{288}}\,{n}^{4}+{\frac {35}{216}}\,{n}^{3}+{
\frac {1}{1440}}\,{n}^{5}+\sum _{\alpha={\rm RootOf} \left( {{\rm \_Z}}
^{2}+{\rm \_Z}+1 \right) }{\frac {1}{81}}\, \left( -1+\alpha \right) {
\alpha}^{-n-1}
other formats
3.4. Asymptotics
4. Ordinary generating function
\displaystyle
{\frac {1}{ \left( -1+x \right) ^{3} \left( -1+{x}^{2} \right) ^{2}
\left( -1+{x}^{3} \right) }}
other formats
5. References
6. Random structure
Search a combinatorial structure by: (firstTerms should be a sequence of integers, separated by commas).