Could anyone please help me evaluate the following sum of squares of binomial coefficients: $$ \sum_{\ell=k}^n \begin{pmatrix} \ell \\ k \end{pmatrix}^2, \quad \mbox{where} \quad 0\leq k \leq n, $$ or, at least, get an upper bound for it? Any suggestion is greatly appreciated

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    $\begingroup$ $k$ is some constant? $\endgroup$ – Shuri2060 Jul 25 '17 at 17:15
  • $\begingroup$ Any more context on the upper bound you need? For instance, is this the runtime of some algorithm? $\endgroup$ – Aryabhata Jul 25 '17 at 17:28
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    $\begingroup$ Just an idea. Perhaps someone can make use of it. $\binom{L}{K}^2$ is the coefficient of $(xy)^K$ in $(1+x)^L(1+y)^L$. $\endgroup$ – Aryabhata Jul 25 '17 at 18:09
  • $\begingroup$ Surely: I am trying to determine if the series $$\sum_{k=0}^{\infty} t^{2k} \sum_{m=0}^k \frac{(-1)^{m+1} 2^m}{(2m)!} r^{2m} \sum_{\ell=m}^{k-m} \ell^2 \ldots (\ell-m+1)^2 r^{2\ell}, \quad |t| <1, \quad |r| < 1, $$ converges or not and thought that the above sum might help me with that... $\endgroup$ – David Lopez Jul 25 '17 at 23:26

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