Central binomial coefficient
They are called central since they show up exactly in the middle of the even-numbered rows in Pascal's triangle. The first few central binomial coefficients starting at n = 0 are:
These numbers have the generating function
The Wallis product can be written in form of an asymptotic for the central binomial coefficient:
The latter can also be easily established by means of Stirling's formula. On the other hand, it can also be used as a means to determine the constant in front of the Stirling formula, by comparison.
Simple bounds are given by
Some better bounds are
and, if more accuracy is required,
- for all
The closely related Catalan numbers Cn are given by:
A slight generalization of central binomial coefficients is to take them as and so the former definition is a particular case when m = 2n, that is, when m is even.
- Koshy, Thomas (2008), Catalan Numbers with Applications, Oxford University Press, ISBN 978-0-19533-454-8.
- Central binomial coefficient at PlanetMath.org.
- Binomial coefficient at PlanetMath.org.
- Pascal's triangle at PlanetMath.org.
- Catalan numbers at PlanetMath.org.