Central Binomial Coefficient - Properties

Properties

These numbers have the generating function


The Wallis product can be written in form of an asymptotics 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 mean 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:

C_n = \frac{1}{n+1} {2n \choose n} = {2n \choose n} - {2n \choose n+1}\text{ for all }n \geq 0.

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.

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