  We show that double sums of the form $$ \sum_{i,j=-n} ^{n}
|i^sj^t(i^k-j^k)^\beta| \binom {2n} {n+i} \binom {2n} {n+j} $$ can always be
expressed in terms of a linear combination of just four functions, namely
$\binom {4n}{2n}$, ${\binom {2n}n}^2$, $4^n\binom {2n}n$, and $16^n$, with
coefficients that are rational in $n$. We provide two different proofs: one is
algorithmic and uses the second author's computer algebra package Sigma; the
second is based on complex contour integrals. In many instances, these results
are extended to double sums of the above form where $\binom {2n}{n+j}$ is
replaced by $\binom {2m}{m+j}$ with independent parameter $m$.
