 | ECS #834: 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) ,C={\rm Set} \left( Z \right) ,
S={\rm Union} \left( B,C \right) \right\}
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3. Coefficients
3.1. First terms
3.2. Recurrence
\displaystyle
\left\{ \left( {n}^{2}+2\,n+1 \right) f \left( n+1 \right) + \left( -
3\,n-{n}^{2}-1 \right) f \left( n+2 \right) +f \left( n+3 \right) n=0,f
\left( 0 \right) =1,f \left( 1 \right) =2,f \left( 2 \right) =2,f
\left( 3 \right) =3,f \left( 4 \right) =7 \right\}
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3.3. Closed form
\displaystyle
1+\Gamma \left( n \right)
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3.4. Asymptotics
4. Exponential generating function
\displaystyle
\ln \left( - \left( -1+x \right) ^{-1} \right) +{{\rm e}^{x}}
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It satisfies the following differential equation of order 2:
\displaystyle
\left\{ \left( -1+2\,x-{x}^{2} \right) {\frac {d}{dx}}y \left( x
\right) + \left( 1-2\,x+{x}^{2} \right) {\frac {d^{2}}{d{x}^{2}}}y
\left( x \right) -x=0,y \left( 0 \right) =1,\mbox {D} \left( y
\right) \left( 0 \right) =2 \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).