In mathematics the nth central binomial coefficient is defined in terms of the binomial coefficient by
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:
- 1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, … (sequence A000984 in the OEIS)
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 only central binomial coefficient that is odd is 1.
The closely related Catalan numbers Cn are given by:
A slight generalization of central binomial coefficients is to take them as , with appropriate real numbers n, where is Gamma function and is Beta function.
The powers of two that divide the central binomial coefficients are given by Gould's sequence.
- Koshy, Thomas (2008), Catalan Numbers with Applications, Oxford University Press, ISBN 978-0-19533-454-8.
This article incorporates material from Central binomial coefficient on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.