sum of powers of binomial coefficients

Some results exist on sums of powers of binomial coefficients. Define As as follows:


For s=1, the binomial theoremMathworldPlanetmath implies that the sum A1(n) is simply 2n.

For s=2, the following result on the sum of the squares of the binomial coefficientsDlmfDlmfMathworldPlanetmath (ni) holds:


that is, A2(n) is the nth central binomial coefficientMathworldPlanetmath.

Proof: This result follows immediately from the Vandermonde identityMathworldPlanetmath:


upon choosing p=q=k=n and observing that (nn-i)=(ni).

Expressions for As(n) for larger values of s exist in terms of hypergeometric functionsDlmfDlmfDlmfMathworldPlanetmath.

Title sum of powers of binomial coefficients
Canonical name SumOfPowersOfBinomialCoefficients
Date of creation 2013-03-22 14:25:43
Last modified on 2013-03-22 14:25:43
Owner Andrea Ambrosio (7332)
Last modified by Andrea Ambrosio (7332)
Numerical id 7
Author Andrea Ambrosio (7332)
Entry type Result
Classification msc 05A10
Classification msc 11B65