 | ECS #23: involutions |
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1. Description
involution without fixed points of n elements even numbers are double factorials
2. Specification
This labelled structure is specified as S in
\displaystyle
\left\{ S={\rm Set} \left( {\rm Cycle} \left( Z,{\rm card}=2 \right)
\right) \right\}
other formats
3. Coefficients
3.1. First terms
3.2. Recurrence
\displaystyle
\left\{ \left( -n-1 \right) f \left( n \right) +f \left( n+2 \right)
=0,f \left( 0 \right) =1,f \left( 1 \right) =0 \right\}
other formats
3.3. Closed form
\displaystyle
\cases{{\frac {{2}^{1/2\,n}\Gamma \left( 1/2\,n+1/2 \right) }{\sqrt {\pi }}}&$n :: {\rm even}$\cr 0&$n :: {\rm odd}$\cr}
other formats
3.4. Asymptotics
4. Exponential generating function
\displaystyle
{{\rm e}^{1/2\,{x}^{2}}}
other formats
5. References
- even number
- EIS A001147
6. Random structure
Search a combinatorial structure by: (firstTerms should be a sequence of integers, separated by commas).