 | ECS #701: A simple grammar |
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1. Description
No description available
2. Specification
This labelled structure is specified as S in
\displaystyle
\left\{ B={\rm Cycle} \left( Z \right) ,S={\rm Prod} \left( Z,B,B
\right) \right\}
other formats
3. Coefficients
3.1. First terms
3.2. Recurrence
\displaystyle
\left\{ \left( {n}^{3}-3\,{n}^{2}-n+{n}^{4}+2 \right) f \left( n
\right) + \left( -2\,{n}^{3}-3\,{n}^{2}+2\,n \right) f \left( n+1
\right) + \left( n+{n}^{2} \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) =6,f \left( 4 \right) =24 \right\}
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3.3. Closed form
\displaystyle
2\,\Gamma \left( n-1 \right) n \left( \Psi \left( n-1 \right) +\gamma
\right)
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3.4. Asymptotics
4. Exponential generating function
\displaystyle
x \left( \ln \left( - \left( x-1 \right) ^{-1} \right) \right) ^{2}
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It satisfies the following differential equation of order 2:
\displaystyle
\left\{ \left( {x}^{2}-3\,x+2 \right) y \left( x \right) + \left( -{x
}^{3}+3\,{x}^{2}-2\,x \right) {\frac {d}{dx}}y \left( x \right) +
\left( {x}^{4}-2\,{x}^{3}+{x}^{2} \right) {\frac {d^{2}}{d{x}^{2}}}y
\left( x \right) -2\,{x}^{3}=0,\mbox {D} \left( y \right) \left( 0
\right) =0, \left( D^{ \left( 2 \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).