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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 $g_{k}$ ?,

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    Yes, thanks, I have corrected my question2012-03-31
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    Write $\sum_k\frac{g_k}{k+1}$ as the integral of a product of two polynomials, then use a substitution. You will recognize an equality involving Gamma and Beta functions.2012-03-31
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    The formula is wrong (try $n=1$, $m=2$).2012-03-31
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    @FelixMarin Your edit makes the indices in the first summation nearly unreadable. Is it necessary? I suggest to slow down on the sophistication of the LaTeX encoding (Knuth would be horrified by the... things you use) and to concentrate on the readability of the end result.2014-07-04

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