# 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.

Title sum of powers of binomial coefficients SumOfPowersOfBinomialCoefficients 2013-03-22 14:25:43 2013-03-22 14:25:43 Andrea Ambrosio (7332) Andrea Ambrosio (7332) 7 Andrea Ambrosio (7332) Result msc 05A10 msc 11B65