Central binomial coefficient

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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:

1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, … (sequence A000984 in the OEIS)

Properties[edit]

These numbers have the generating function

The Wallis product can be written in form of an asymptotic 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 in front of the Stirling formula, by comparison.

Simple bounds are given by

[citation needed]

Some better bounds are

[citation needed]

and, if more accuracy is required,

for all [citation needed]

The only central binomial coefficient that is odd is 1.[citation needed]

Related sequences[edit]

The closely related Catalan numbers Cn are given by:

A slight generalization of central binomial coefficients is to take them as , with appropriate real numbers n, where is Gamma function and is Beta function.

See also[edit]

References[edit]

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

External links[edit]

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