Central binomial coefficient

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.

Related sequences

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.

See also


    External links

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