  We consider several families of binomial sum identities whose definition
involves the absolute value function. In particular, we consider centered
double sums of the form \[S_{\alpha,\beta}(n) :=
\sum_{k,\;\ell}\binom{2n}{n+k}\binom{2n}{n+\ell}
|k^\alpha-\ell^\alpha|^\beta,\] obtaining new results in the cases $\alpha = 1,
2$. We show that there is a close connection between these double sums in the
case $\alpha=1$ and the single centered binomial sums considered by Tuenter.
