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Just wandering if there are any criteria that can decide whether a finite series summation has closed form or not. for example, In the following nested summation, $n$ is some even integer that will be specified.

$\begin{equation} f(n) =\sum_{k=0}^{n/2}\sum_{l=0}^{n/2}\sum_{i=0}^{k}\sum_{j=0}^{l}\frac{1}{n-k-l}\frac{1}{k+l}\binom{n/2}{k-i}\binom{n/2-(k-i)}{2i}\binom{n/2}{l-j}\binom{n/2-(l-j)}{2j}\binom{2i+2j}{i+j} \end{equation}$

Of course, the cases of $l=k=0$ and $l=k=n/2$ are excluded from the summation. The question is, do we have any criteria to judge whether a complicated finite series summation has closed form expression or not? or it is because the problem is simple and we are lucky that we can find one closed form for complicated summations. most of the time, we are not able to find one.

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  • $\begingroup$ Maybe this in not a correctly phrased question. A summation is like an integration in some sense, and a closed form expression for some multiple-variable integration can not be worked out. $\endgroup$ – Ming Li Aug 12 '15 at 19:30
  • $\begingroup$ One good start would be to compute it for small $n$ and see if you recognize it. $\endgroup$ – Will Sawin Aug 12 '15 at 19:41
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Your sum does not appear to have a closed form. However, everything you ever wanted to know about summations like this can be found in "A=B" by Petkovsek, Wilf, and Zeilberger.

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