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 $s$ a positive integer and $n$ a nonnegative integer.

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.