Some binomial sums involving absolute values

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 $$S_{\alpha ,\beta }(n):=\sum _{k,\mathrm{\ell }}\left(\genfrac{}{}{0px}{}{2n}{n+k}\right)\left(\genfrac{}{}{0px}{}{2n}{n+\mathrm{\ell }}\right)\mathrm{\mid }{k}^{\alpha }-{\mathrm{\ell }}^{\alpha }{\mathrm{\mid }}^{\beta },$$ 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.