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:

The central binomial coefficients have ordinary generating function

and exponential generating function

where I₀ is a modified Bessel function of the first kind.

They also satisfy the recurrence

The Wallis product can be written in asymptotic form 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 ${\displaystyle {\sqrt {2\pi }}}$ in front of the Stirling formula, by comparison.

Simple bounds are given by

Some better bounds are

and, if more accuracy is required,

The only central binomial coefficient that is odd is 1.

The closely related Catalan numbers C_n are given by:

A slight generalization of central binomial coefficients is to take them as ${\displaystyle {\frac {\Gamma (2n+1)}{\Gamma (n+1)^{2}}}={\frac {1}{n\mathrm {B} (n+1,n)}}}$, with appropriate real numbers n, where ${\displaystyle \Gamma (x)}$ is the Gamma function and ${\displaystyle \mathrm {B} (x,y)}$ is the Beta function.

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