The sum $ g_{k}=\sum_{\vphantom{\LARGE A}j,\,i \atop {\vphantom{\LARGE A}i\ +\ j\ =\ k}} \left(-1\right)^{\,j}{m \choose j}{n \choose i},\qquad \mbox{for}\quad 0 \leq j \leq m\quad\mbox{and}\quad 0 \leq i \leq n, $ is involved in the development of the polynomial $(1+x)^n(1-x)^m$.
It seems to me that the sum $\sum_{k}{g_{k} \over k+1} = {2^{n + m} \over (n+m+1){n+m \choose n}}. $ Could anybody provide a formal proof or a closed-form development of $g_{k}$ ?,