$\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle} \newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack} \newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,} \newcommand{\dd}{{\rm d}} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,{\rm e}^{#1}\,} \newcommand{\fermi}{\,{\rm f}} \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,} \newcommand{\half}{{1 \over 2}} \newcommand{\ic}{{\rm i}} \newcommand{\iff}{\Longleftrightarrow} \newcommand{\imp}{\Longrightarrow} \newcommand{\pars}[1]{\left(\, #1 \,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\pp}{{\cal P}} \newcommand{\root}[2][]{\,\sqrt[#1]{\vphantom{\large A}\,#2\,}\,} \newcommand{\sech}{\,{\rm sech}} \newcommand{\sgn}{\,{\rm sgn}} \newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}} \newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert}$ $\ds{\sum_{m = 0}^{n}\pars{-1}^{n - m}{n \choose m}{m - 1 \choose \ell}:\ {\large ?}.\qquad\ell \geq 0}$
\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}