 | ECS #71: Euler's counting of triangulations |
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1. Description
- Euler's counting of triangulations: number of ways of cutting up a convex(n+2)-gon into n triangles by means of (n-1) non-intersecting diagonals
- Catalan numbers
2. Specification
This unlabelled structure is specified as S in
\displaystyle
\left\{ S={\rm Union} \left( Z,{\rm Prod} \left( U,Z,S \right) ,{\rm
Prod} \left( S,Z,U \right) ,{\rm Prod} \left( S,Z,S \right) \right) ,U
={\rm Epsilon} \right\}
other formats
3. Coefficients
3.1. First terms
3.2. Recurrence
\displaystyle
\left\{ \left( 2+4\,n \right) f \left( n \right) + \left( -2-n
\right) f \left( n+1 \right) =0,f \left( 0 \right) =0,f \left( 1
\right) =1 \right\}
other formats
3.3. Closed form
\displaystyle
4\,{\frac {{2}^{2\,n-2}\Gamma \left( n+1/2 \right) }{\sqrt {\pi }
\Gamma \left( 2+n \right) }}
other formats
3.4. Asymptotics
4. Ordinary generating function
\displaystyle
-1/2\,{\frac {-1+2\,x+\sqrt {1-4\,x}}{x}}
other formats
5. References
- Euler, 1753
- EIS A000108
6. Random structure
Search a combinatorial structure by: (firstTerms should be a sequence of integers, separated by commas).