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Binomial identity of alternating sum of products of binomial coefficients taken two at a time
I came across this identity a while back and wasn't able to prove it.
$$\sum_{i=1}^{n-3}\frac{\binom{n-3}{i}\binom{n+i-1}{i}}{i+1}\cdot(-1)^{i+1}=
\begin{cases}
0& \text{if $n$ is odd,}\\
2& \...
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