05A10-SumOfPowersOfBinomialCoefficients.tex

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\pmtitle{sum of powers of binomial coefficients}
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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.

{\bf 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.
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