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$\ds{\sum_{m = 0}^{n}\pars{-1}^{n - m}{n \choose m}{m - 1 \choose \ell}:\
{\large ?}.\qquad\ell \geq 0}$
\begin{align}
&\color{#66f}{\large\sum_{m = 0}^{n}\pars{-1}^{n - m}{n \choose m}
{m - 1 \choose \ell}}
\\[3mm]&=\pars{-1}^{n}\sum_{m = 0}^{n}\pars{-1}^{m}{n \choose m}
\oint_{0\ <\ \verts{z}\ =\ a\ <\ 1}{\pars{1 + z}^{m - 1} \over z^{\ell + 1}}
\,{\dd z \over 2\pi\ic}
\\[3mm]&=\pars{-1}^{n}\oint_{0\ <\ \verts{z}\ =\ a\ <\ 1}
{1 \over z^{\ell + 1}\pars{1 + z}}
\sum_{m = 0}^{n}{n \choose m}\pars{-z - 1}^{m}\,{\dd z \over 2\pi\ic}
\\[3mm]&=\pars{-1}^{n}\oint_{0\ <\ \verts{z}\ =\ a\ <\ 1}
{1 \over z^{\ell + 1}\pars{1 + z}}
\bracks{1 + \pars{-z - 1}}^{n}\,{\dd z \over 2\pi\ic}
\\[3mm]&=\oint_{0\ <\ \verts{z}\ =\ a\ <\ 1}{1 \over z^{\ell - n + 1}\pars{1 + z}}
{\dd z \over 2\pi\ic}
=\sum_{k = 0}^{\infty}\pars{-1}^{k}\oint_{0\ <\ \verts{z}\ =\ a\ <\ 1}{1 \over z^{\ell - n - k + 1}}{\dd z \over 2\pi\ic}
\\[3mm]&=\sum_{k = 0}^{\infty}\pars{-1}^{k}\,\delta_{\ell - n,k}
=\color{#66f}{\large\left\lbrace\begin{array}{lcl}
\pars{-1}^{\ell - n} & \mbox{if} & \ell \geq n
\\[2mm]
0&&\mbox{otherwise}
\end{array}\right.}
\end{align}