 | ECS #54: Labelled Plane Binary Trees |
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1. Description
Binary Trees of n elements (external nodes or internal nodes)
2. Specification
This labelled structure is specified as S in
\displaystyle
\left\{ S={\rm Union} \left( Z,{\rm Prod} \left( Z,S,S \right)
\right) \right\}
other formats
3. Coefficients
3.1. First terms
3.2. Recurrence
\displaystyle
\left\{ \left( -12\,{n}^{2}-4\,{n}^{3}-8\,n \right) f \left( n
\right) + \left( 3+n \right) f \left( n+2 \right) =0,f \left( 0
\right) =0,f \left( 1 \right) =1 \right\}
other formats
3.3. Closed form
\displaystyle
\cases{0&$n :: {\rm even}$\cr 8\,{\frac {{2}^{2\,n-2} \left( \Gamma \left( 1/2\,n+1 \right) \right) ^{2}}{n\pi \, \left( 1+n \right) }}&$n :: {\rm odd}$\cr}
other formats
3.4. Asymptotics
4. Exponential generating function
\displaystyle
-1/2\,{\frac {-1+\sqrt {1-4\,{x}^{2}}}{x}}
other formats
5. References
6. Random structure
Search a combinatorial structure by: (firstTerms should be a sequence of integers, separated by commas).