central binomial coefficient

The nth central binomial coefficientMathworldPlanetmath is defined to be


where (2nn) is a binomial coefficientMathworldPlanetmath. These numbers have the generating functionMathworldPlanetmath


They are closely related to the Catalan sequence, in that


Alternate definition

A less frequently-encountered definition for the nth central binomial coefficient is (nn2).

Note that the set of these numbers meeting this alternate criterion is a supersetMathworldPlanetmath of those meeting the first criterion, since for n=2m we have


By cancelling terms of one of the n!’s against terms of the 2n!, one may rewrite the central binomial coefficient as follows:


Alternatively, one may cancel each term of the n! against twice itself, leaving 2’s in the numerator:


Doubling the terms in the denominator, we obtain an expression for the central binomial coeficient in terms of a quotient of successive odd numbersMathworldPlanetmathPlanetmath by successive even numbers:


By means of these formulae, one may derive some important properties of the central binomial coeficients. By examining the first two formulae, one may deduce results about the prime factorsMathworldPlanetmathPlanetmath of central binomial coefficients (for proofs, please see the attachments to this entry):

Theorem 1

If n3 is an integer and p is a prime numberMathworldPlanetmath such that n<p<2n, then p divides (2nn).

Theorem 2

If n3 is an integer and p is a prime number such that 2n/3<pn, then p does not divide (2nn).

In conjunctionMathworldPlanetmath with Wallis’ formulaMathworldPlanetmathPlanetmath for π, the third formula for the central binomial coefficient may be used to derive an asymptotic expression, as is done in an attachment to this entry:

Title central binomial coefficient
Canonical name CentralBinomialCoefficient
Date of creation 2013-03-22 14:25:40
Last modified on 2013-03-22 14:25:40
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 8
Author rspuzio (6075)
Entry type Definition
Classification msc 05A10
Classification msc 11B65
Related topic BinomialCoefficient
Related topic CatalanNumbers