 | ECS #750: 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 Set} \left( Z,1\leq {\rm card} \right) ,S={\rm Prod}
\left( Z,Z,Z,Z,B,B \right) \right\}
other formats
3. Coefficients
3.1. First terms
3.2. Recurrence
\displaystyle
\left\{ \left( 2\,{n}^{2}+6\,n+4 \right) f \left( n \right) + \left(
18-3\,{n}^{2}+3\,n \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 \right\}
other formats
3.3. Closed form
\displaystyle
2\,n \left( -1+n \right) \left( n-2 \right) \left( n-3 \right)
\left( {2}^{n-5}-1 \right)
other formats
3.4. Asymptotics
4. Exponential generating function
\displaystyle
{x}^{4} \left( {{\rm e}^{x}} \right) ^{2}-2\,{x}^{4}{{\rm e}^{x}}+{x}^{
4}
other formats
It satisfies the following differential equation of order 2:
\displaystyle
\left\{ \left( 20+12\,x+2\,{x}^{2} \right) y \left( x \right) +
\left( -8\,x-3\,{x}^{2} \right) {\frac {d}{dx}}y \left( x \right) +
\left( {\frac {d^{2}}{d{x}^{2}}}y \left( x \right) \right) {x}^{2}-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\}
other formats
5. References
6. Random structure
Search a combinatorial structure by: (firstTerms should be a sequence of integers, separated by commas).