 | ECS #116 |
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1. Description
No description available
2. Specification
This labelled structure is specified as S in
\displaystyle
\left\{ B={\rm Set} \left( Z \right) ,C={\rm Cycle} \left( Z \right) ,
S={\rm Prod} \left( C,B \right) \right\}
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3. Coefficients
3.1. First terms
3.2. Recurrence
\displaystyle
\left\{ \left( -n-1 \right) f \left( n \right) + \left( 4+2\,n
\right) f \left( n+1 \right) + \left( -n-4 \right) f \left( n+2
\right) +f \left( n+3 \right) =0,f \left( 0 \right) =0,f \left( 1
\right) =1,f \left( 2 \right) =3 \right\}
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3.3. Closed form
\displaystyle
\sum _{{\rm \_n0}=0}^{n-1}{{\rm e}^{1}}\Gamma \left( {\rm \_n0}+1,1
\right)
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3.4. Asymptotics
4. Exponential generating function
\displaystyle
\ln \left( - \left( -1+x \right) ^{-1} \right) {{\rm e}^{x}}
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It satisfies the following differential equation of order 2:
\displaystyle
\left\{ \left( -2+x \right) y \left( x \right) + \left( 3-2\,x
\right) {\frac {d}{dx}}y \left( x \right) + \left( -1+x \right) {
\frac {d^{2}}{d{x}^{2}}}y \left( x \right) =0,y \left( 0 \right) =0,
\mbox {D} \left( y \right) \left( 0 \right) =1 \right\}
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5. References
6. Random structure
Search a combinatorial structure by: (firstTerms should be a sequence of integers, separated by commas).