 | ECS #432 |
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1. Description
No description available
2. Specification
This unlabelled structure is specified as S in
\displaystyle
\left\{ B={\rm Sequence} \left( S \right) ,S={\rm Prod} \left( Z,B,B
\right) \right\}
other formats
3. Coefficients
3.1. First terms
3.2. Recurrence
\displaystyle
\left\{ \left( -3+27\,{n}^{2} \right) f \left( n \right) + \left( -6
\,n-2-4\,{n}^{2} \right) f \left( n+1 \right) =0,f \left( 0 \right) =0,
f \left( 1 \right) =1 \right\}
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3.3. Closed form
\displaystyle
{\frac {{3}^{-3/2+3\,n}\Gamma \left( n+1/3 \right) \Gamma \left( n-1/
3 \right) }{\pi \,\Gamma \left( 2\,n+1 \right) }}
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3.4. Asymptotics
4. Ordinary generating function
\displaystyle
x \left( {\rm RootOf} \left( -{\rm \_Z}+{{\rm \_Z}}^{3}x+1 \right)
\right) ^{2}
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It satisfies the following differential equation of order 2:
\displaystyle
\left\{ -3\,y \left( x \right) + \left( -2+27\,x \right) {\frac {d}{dx
}}y \left( x \right) +2+ \left( 27\,{x}^{2}-4\,x \right) {\frac {d^{2}}
{d{x}^{2}}}y \left( x \right) =0,y \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).