# Central binomial coefficient

In mathematics the *n*th **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:

## Properties

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.

## Related sequences

The closely related Catalan numbers *C*_{n} 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.

## See also

## References

- Koshy, Thomas (2008),
*Catalan Numbers with Applications*, Oxford University Press, ISBN 978-0-19533-454-8.

## External links

- "Central binomial coefficient".
*PlanetMath*. - "Binomial coefficient".
*PlanetMath*. - "Pascal's triangle".
*PlanetMath*. - "Catalan numbers".
*PlanetMath*.

*This article incorporates material from Central binomial coefficient on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.*