 | ECS #76: Ways of parenthezing a product |
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1. Description
- Number of ways of completely parenthesizing a product of n letters
- Catalan numbers
2. Specification
This unlabelled structure is specified as S in
\displaystyle
\left\{ S={\rm Union} \left( {\rm Prod} \left( Z,Z \right) ,{\rm Prod}
\left( Z,S \right) ,{\rm Prod} \left( S,Z \right) ,{\rm Prod} \left( S
,S \right) \right) \right\}
other formats
3. Coefficients
3.1. First terms
3.2. Recurrence
\displaystyle
\left\{ \left( -2+4\,n \right) f \left( n \right) + \left( -n-1
\right) f \left( n+1 \right) =0,f \left( 0 \right) =0,f \left( 1
\right) =0,f \left( 2 \right) =1 \right\}
other formats
3.3. Closed form
\displaystyle
4\,{\frac {{2}^{2\,n-4}\Gamma \left( n-1/2 \right) }{\sqrt {\pi }
\Gamma \left( n+1 \right) }}
other formats
3.4. Asymptotics
4. Ordinary generating function
\displaystyle
1/2-x-1/2\,\sqrt {1-4\,x}
other formats
5. References
- EIS A000108
- http://www.astro.virginia.edu/~eww6n/math/CatalanNumber.html
- http://forum.swarthmore.edu/advanced/robertd/catalan.html
6. Random structure
Search a combinatorial structure by: (firstTerms should be a sequence of integers, separated by commas).