 | ECS #85: Pairs of cycles |
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1. Description
Pairs of cycles of cardinality at least 3
2. Specification
This labelled structure is specified as S in
\displaystyle
\left\{ B={\rm Cycle} \left( Z,3\leq {\rm card} \right) ,S={\rm Prod}
\left( B,B \right) \right\}
other formats
3. Coefficients
3.1. First terms
3.2. Recurrence
\displaystyle
\left\{ \left( -{n}^{2}+{n}^{3}-2\,n \right) f \left( n \right) +
\left( -2\,{n}^{2}+n+3 \right) f \left( n+1 \right) + \left( n-1
\right) f \left( n+2 \right) =0,f \left( 0 \right) =0,f \left( 1
\right) =0,f \left( 2 \right) =0,f \left( 3 \right) =0,f \left( 4
\right) =0,f \left( 5 \right) =0,f \left( 6 \right) =80,f \left( 7
\right) =840 \right\}
other formats
3.3. Closed form
\displaystyle
\Gamma \left( -2+n \right) \left( n-1 \right) \left( -2+n \right)
\left( -3+2\,\Psi \left( -2+n \right) +2\,\gamma \right)
other formats
3.4. Asymptotics
4. Exponential generating function
\displaystyle
\left( \ln \left( - \left( -1+x \right) ^{-1} \right) \right) ^{2}-2
\,\ln \left( - \left( -1+x \right) ^{-1} \right) x-\ln \left( -
\left( -1+x \right) ^{-1} \right) {x}^{2}+{x}^{2}+{x}^{3}+1/4\,{x}^{4}
other formats
It satisfies the following differential equation of order 2:
\displaystyle
\left\{ \left( -2+3\,x-{x}^{2} \right) {\frac {d}{dx}}y \left( x
\right) + \left( x-2\,{x}^{2}+{x}^{3} \right) {\frac {d^{2}}{d{x}^{2}}
}y \left( x \right) -2\,{x}^{5}=0,y \left( 0 \right) =0, \left( D^{
\left( 3 \right) } \right) \left( y \right) \left( 0 \right) =0
\right\}
other formats
5. References
6. Random structure
Search a combinatorial structure by: (firstTerms should be a sequence of integers, separated by commas).