Central binomial coefficient

In mathematics the nth central binomial coefficient is defined in terms of the binomial coefficient by

{2n \choose n} = \frac{(2n)!}{(n!)^2}\text{ for all }n \geq 0.

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 OEIS)

Properties

These numbers have the generating function

\frac{1}{\sqrt{1-4x}} = 1 + 2x + 6x^2 + 20x^3 + 70x^4 + 252x^5 + \cdots.

The Wallis product can be written in form of an asymptotic for the central binomial coefficient:

 {2n \choose n} \sim \frac{4^n}{\sqrt{\pi n}}\text{ as }n\rightarrow\infty.

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 \sqrt{2\pi} in front of the Stirling formula, by comparison.

Simple bounds are given by

\frac{4^n}{2n+1} \leq {2n \choose n} \leq 4^n\text{ for all }n \geq 1

Some better bounds are

\frac{4^n}{\sqrt{4n}} \leq {2n \choose n} \leq \frac{4^n}{\sqrt{3n+1}}\text{ for all }n \geq 1

and, if more accuracy is required,

{2n \choose n} = \frac{4^n}{\sqrt{\pi n}}\left(1-\frac{c_n}{n}\right)\text{ where }\frac{1}{9} < c_n < \frac{1}{8} for all n \geq 1.

Related sequences

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  \frac{\Gamma(2n+1)}{\Gamma(n+1)^2}=\frac{1}{n \Beta(n+1,n)}, with appropriate real numbers n, where \Gamma(x) is Gamma function and \Beta(x,y) is Beta function.

See also

References

External links

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