sum of powers of binomial coefficients

Some results exist on sums of powers of binomial coefficients. Define $A_{s}$ as follows:

$A_{s}(n)=\sum_{i=0}^{n}{n\choose i}^{s}$

For $s=1$ , the binomial theorem implies that the sum $A_{1}(n)$ is simply $2^{n}$ .

For $s=2$ , the following result on the sum of the squares of the binomial coefficients ${n\choose i}$ holds:

$A_{2}(n)=\sum_{i=0}^{n}{n\choose i}^{2}={2n\choose n}$

that is, $A_{2}(n)$ is the $n$ th central binomial coefficient.

Proof: This result follows immediately from the Vandermonde identity:

${p+q\choose k}=\sum_{i=0}^{k}{p\choose i}{q\choose k-i}$

upon choosing $p=q=k=n$ and observing that ${n\choose n-i}={n\choose i}$ .

Expressions for $A_{s}(n)$ for larger values of $s$ exist in terms of hypergeometric functions.

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