 | ECS #353: Integer partition |
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1. Description
Partition of n in at most 4 parts
2. Specification
This unlabelled structure is specified as S in
\displaystyle
\left\{ S={\rm Set} \left( {\rm Sequence} \left( Z,1\leq {\rm card}
\right) ,{\rm card}\leq 4 \right) \right\}
other formats
3. Coefficients
3.1. First terms
3.2. Recurrence
\displaystyle
\left\{ 6\,f \left( n \right) +18\,f \left( n+1 \right) +30\,f \left(
n+2 \right) +36\,f \left( n+3 \right) +30\,f \left( n+4 \right) +18\,f
\left( n+5 \right) +6\,f \left( n+6 \right) -{n}^{3}-24\,{n}^{2}-191\,
n-504=0,f \left( 0 \right) =1,f \left( 1 \right) =1,f \left( 2 \right)
=2,f \left( 3 \right) =3,f \left( 4 \right) =5,f \left( 5 \right) =6
\right\}
other formats
3.3. Closed form
\displaystyle
{\frac {175}{288}}+\sum _{\alpha={\rm RootOf} \left( {{\rm \_Z}}^{2}+{
\rm \_Z}+1 \right) }1/27\, \left( -1+\alpha \right) {\alpha}^{-1-n}+{
\frac {5}{48}}\,{n}^{2}+{\frac {15}{32}}\,n+{\frac {1}{144}}\,{n}^{3}+
\sum _{\alpha={\rm RootOf} \left( {{\rm \_Z}}^{2}+1 \right) }1/16\,{
\alpha}^{-n}+{\frac {5}{32}}\, \left( -1 \right) ^{-n}+1/32\, \left( -1
\right) ^{-n}n
other formats
3.4. Asymptotics
4. Ordinary generating function
\displaystyle
{\frac {1}{ \left( {x}^{2}+x+1 \right) \left( -1+{x}^{2} \right)
\left( x+1 \right) \left( -1+x \right) ^{3} \left( {x}^{2}+1 \right)
}}
other formats
5. References
6. Random structure
Search a combinatorial structure by: (firstTerms should be a sequence of integers, separated by commas).