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 $n$ 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.