 | ECS #856: 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( C \right) ,C={\rm Sequence} \left( Z,1
\leq {\rm card} \right) ,S={\rm Prod} \left( B,B \right) \right\}
other formats
3. Coefficients
3.1. First terms
3.2. Recurrence
\displaystyle
\left\{ \left( 4\,{n}^{4}+16\,{n}^{3}+20\,{n}^{2}+8\,n \right) f
\left( n \right) + \left( -12\,{n}^{3}-54\,{n}^{2}-78\,n-36 \right) f
\left( n+1 \right) + \left( 13\,{n}^{2}+52+52\,n \right) f \left( n+2
\right) + \left( -15-6\,n \right) f \left( n+3 \right) +f \left( n+4
\right) =0,f \left( 0 \right) =0,f \left( 1 \right) =0,f \left( 2
\right) =2,f \left( 3 \right) =18 \right\}
other formats
3.3. Asymptotics
4. Exponential generating function
\displaystyle
\left( \ln \left( {\frac {-1+x}{-1+2\,x}} \right) \right) ^{2}
other formats
It satisfies the following differential equation of order 2:
\displaystyle
\left\{ \left( -3+13\,x-18\,{x}^{2}+8\,{x}^{3} \right) {\frac {d}{dx}
}y \left( x \right) -2+ \left( 1-6\,x+13\,{x}^{2}-12\,{x}^{3}+4\,{x}^{4
} \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) =0 \right\}
other formats
5. References
6. Random structure
Search a combinatorial structure by: (firstTerms should be a sequence of integers, separated by commas).