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 the 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 $n\geq 1.$

The only central binomial coefficient that is odd is 1.

Related sequences

The closely related Catalan numbers C_n 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 ${\frac {\Gamma (2n+1)}{\Gamma (n+1)^{2}}}={\frac {1}{n\mathrm {B} (n+1,n)}}$, with appropriate real numbers n, where $\Gamma (x)$ is Gamma function and $\mathrm {B} (x,y)$ is Beta function.

See also

• Erdős squarefree conjecture
• Central multinomial coefficients

References

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

External links

• "Central binomial coefficient". PlanetMath.
• "Binomial coefficient". PlanetMath.
• "Pascal's triangle". PlanetMath.
• "Catalan numbers". PlanetMath.

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