TITLE: A Binomial Identity simplify
QUESTION [21 upvotes]: I want to symplify
$$ \sum_{\ell=1}^{k} \frac{1}{\ell}\sum_{m=1}^{\min\{\ell,k-\ell\}}\binom{\ell}{m}\binom{k-\ell-1}{m-1}.
$$

REPLY [2 votes]: By way of an extended comment in response to a pers. comm. / request. A conjectured alternate representation of the sum (here $i=p$ and $j=q$) is given by
$$k (-1)^p \mathrm{Res}_{z=0} z^{p-1}
[w^{p+q}] (1+w)^{k}
[v^{k}] (1+v)^{k-1} \\ \times
\sum_{\ell\ge 1} \frac{(-1)^\ell}{\ell}
z^{-2\ell} w^\ell (1+w)^{-2\ell}
v^\ell (1+v)^{-\ell}
(z(1+w)^2+v(1+z)(z+w^2))^\ell.$$
At this point the term in $1/\ell$ introduces a logarithm and the power term in $\ell$ does not factorize / collect easily in $z$ or $w.$