 | ECS #109 |
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1. Description
No description available
2. Specification
This labelled structure is specified as S in
\displaystyle
\left\{ B={\rm Prod} \left( C,Z \right) ,C={\rm Sequence} \left( B
\right) ,S={\rm Cycle} \left( B \right) \right\}
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3. Coefficients
3.1. First terms
3.2. Recurrence
\displaystyle
\left\{ \left( -2\,n-4\,{n}^{2} \right) f \left( n \right) + \left( n
+1 \right) f \left( n+1 \right) =0,f \left( 0 \right) =0,f \left( 1
\right) =1 \right\}
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3.3. Closed form
\displaystyle
{\frac {{2}^{-1+2\,n}\Gamma \left( n+1/2 \right) }{\sqrt {\pi }n}}
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3.4. Asymptotics
4. Exponential generating function
\displaystyle
\ln \left( - \left( -1/2-1/2\,\sqrt {1-4\,x} \right) ^{-1} \right)
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It satisfies the following differential equation of order 2:
\displaystyle
\left\{ \left( 6\,x-1 \right) {\frac {d}{dx}}y \left( x \right) +1+
\left( -x+4\,{x}^{2} \right) {\frac {d^{2}}{d{x}^{2}}}y \left( x
\right) =0,y \left( 0 \right) =0 \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).