 | ECS #757: 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,Z,Z,Z,B,
B \right) \right\}
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3. Coefficients
3.1. First terms
3.2. Recurrence
\displaystyle
\left\{ \left( {n}^{4}-5\,{n}^{3}-6\,{n}^{2}+32\,n+32 \right) f
\left( n \right) + \left( -2\,{n}^{3}+9\,{n}^{2}+5\,n-42 \right) f
\left( n+1 \right) + \left( 6-5\,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) =0,f \left( 4 \right) =0,f \left( 5
\right) =0,f \left( 6 \right) =720,f \left( 7 \right) =5040 \right\}
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3.3. Closed form
\displaystyle
2\,\Gamma \left( n-4 \right) n \left( n-1 \right) \left( n-2 \right)
\left( n-3 \right) \left( \Psi \left( n-4 \right) +\gamma \right)
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3.4. Asymptotics
4. Exponential generating function
\displaystyle
{x}^{4} \left( \ln \left( - \left( -1+x \right) ^{-1} \right)
\right) ^{2}
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It satisfies the following differential equation of order 2:
\displaystyle
\left\{ \left( 20-36\,x+16\,{x}^{2} \right) y \left( x \right) +
\left( -8\,x+15\,{x}^{2}-7\,{x}^{3} \right) {\frac {d}{dx}}y \left( x
\right) + \left( {x}^{2}-2\,{x}^{3}+{x}^{4} \right) {\frac {d^{2}}{d{x
}^{2}}}y \left( x \right) -2\,{x}^{6}=0, \left( D^{ \left( 4 \right) }
\right) \left( y \right) \left( 0 \right) =0, \left( D^{ \left( 5
\right) } \right) \left( y \right) \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).