 | ECS #743: 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,B,B,
B \right) \right\}
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3. Coefficients
3.1. First terms
3.2. Recurrence
\displaystyle
\left\{ \left( -42\,{n}^{3}+135\,n+16\,{n}^{4}+162-81\,{n}^{2}+3\,{n}
^{5}-{n}^{6} \right) f \left( n \right) + \left( -228-38\,{n}^{3}+81\,{
n}^{2}+104\,n-6\,{n}^{4}+3\,{n}^{5} \right) f \left( n+1 \right) +
\left( 6\,{n}^{3}+21\,{n}^{2}+36-60\,n-3\,{n}^{4} \right) f \left( n+2
\right) + \left( {n}^{3}-3\,{n}^{2}+2\,n \right) f \left( n+3 \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) =7560 \right\}
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3.3. Closed form
\displaystyle
\Gamma \left( n-3 \right) n \left( n-1 \right) \left( n-2 \right)
\left( 6\,\gamma\,\Psi \left( n-3 \right) -6\,\gamma+6\,{\gamma}^{2}+
\sum _{{\rm \_n0}=0}^{n-6}6\,{\frac {\Psi \left( {\rm \_n0}+2 \right) }
{{\rm \_n0}+2}} \right)
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3.4. Asymptotics
4. Exponential generating function
\displaystyle
{x}^{3} \left( \ln \left( - \left( -1+x \right) ^{-1} \right)
\right) ^{3}
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It satisfies the following differential equation of order 3:
\displaystyle
\left\{ \left( 60+111\,{x}^{2}-27\,{x}^{3}-144\,x \right) y \left( x
\right) + \left( 19\,{x}^{4}-73\,{x}^{3}+90\,{x}^{2}-36\,x \right) {
\frac {d}{dx}}y \left( x \right) + \left( -6\,{x}^{5}-24\,{x}^{3}+21\,{
x}^{4}+9\,{x}^{2} \right) {\frac {d^{2}}{d{x}^{2}}}y \left( x \right) +
\left( 3\,{x}^{4}-3\,{x}^{5}+{x}^{6}-{x}^{3} \right) {\frac {d^{3}}{d{
x}^{3}}}y \left( x \right) +6\,{x}^{6}=0, \left( D^{ \left( 3 \right) }
\right) \left( y \right) \left( 0 \right) =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).