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.

The powers of two that divide the central binomial coefficients are given by Gould's sequence.

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.

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Central binomial coefficient

Properties

Related sequences

See also

References

External links