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In mathematics the *n*th **central binomial coefficient** is the particular 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:

The central binomial coefficients represent the number of combinations of a set where there are an equal number of two types of objects.

For example, represents *AABB, ABAB, ABBA, BAAB, BABA, BBAA*.

They also represent the number of combinations of *A* and *B* where there are never more *B* 's than *A* 's.

For example, represents *AAAA, AAAB, AABA, AABB, ABAA, ABAB*.

The number of factors of *2* in is equal to the number of ones in the binary representation of *n*,^{[1]} so *1* is the only odd central binomial coefficient.

The ordinary generating function for the central binomial coefficients is

This can be proved using the binomial series and the relation

where is a generalized binomial coefficient.

The central binomial coefficients have exponential generating function

where

The generating function of the squares of the central binomial coefficients can be written in terms of the complete elliptic integral of the first kind:^{[citation needed]}

The Wallis product can be written using limits:

because .

Taking the square root of both sides gives the asymptote for the central binomial coefficient:

- .

The latter can also be 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.

Simple bounds that immediately follow from are

Some better bounds are

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 the gamma function and is the beta function.

The powers of two that divide the central binomial coefficients are given by Gould's sequence, whose *n*th element is the number of odd integers in row *n* of Pascal's triangle.

Squaring the generating function gives

Comparing the coefficients of gives

Half the central binomial coefficient (for ) (sequence A001700 in the OEIS) is seen in Wolstenholme's theorem.

By the Erdős squarefree conjecture, proved in 1996, no central binomial coefficient with *n* > 4 is squarefree.

is the sum of the squares of the *n*-th row of Pascal's Triangle:^{[3]}

For example, .

Erdős uses central binomial coefficients extensively in his proof of Bertrand's postulate.

Another noteworthy fact is that the power of 2 dividing is exactly n.

**^**Sloane, N. J. A. (ed.). "Sequence A000120".*The On-Line Encyclopedia of Integer Sequences*. OEIS Foundation.**^**Stanley, Richard P. (2012),*Enumerative Combinatorics*,**1**(2 ed.), Cambridge University Press, Example 1.1.15, ISBN 978-1-107-60262-5- ^
^{a}^{b}Sloane, N. J. A. (ed.). "Sequence A000984 (Central binomial coefficients)".*The On-Line Encyclopedia of Integer Sequences*. OEIS Foundation.

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

- Central binomial coefficient at PlanetMath.