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got the following problem to prove for $n \in \mathbb{N}$ and $1 \leq i \leq n$:

\begin{equation} \sum_{k=1}^{n-i} \frac{(2n-1-i-2k)! 2^{2k}}{(n-i-k)! (n-k)! 2}=\sum_{k=1}^{n-i} \frac{(2n-1-i-k)! 2^{k}}{(n-i-k)! (n-1)! i} \end{equation}

Numerical tests in Maple say it is correct. I tried to work out explicit solutions for both sums which did not work. Again Maple does not give an explicit solution when using the "simplify" button (which of course does not mean anything). Also played around with the sums but did not get sth useful. Obviously, the summands on the lhs and rhs are also not the same. Do you have an idea how to tackle this problem? Thank you

Michael Hardy
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schnubert
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  • I have an idea. Both sums may be equal to a coefficient of a polynomial (see, for instance, my answer to a similar question). In your case you maybe need to differentiate or integrate binomial powers in order to obtain suitable polynomials. – Alex Ravsky Jan 30 '15 at 04:00

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