I am trying to find the value of: $$\sum_k (-1)^k {n \choose k}^2$$ The hint is that $(1-x^2)^n = (1-x)^n(1+x)^n$
I tried solving it using convolution but it didn't quite work.
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E. Joseph
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user1435820
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Ow, sorry. That was a typo – user1435820 Jun 27 '17 at 12:22
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is this a finite sum? – Dr. Sonnhard Graubner Jun 27 '17 at 12:23
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It's summing over all non-negative integers – user1435820 Jun 27 '17 at 12:24
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your sum containes the Gamma-function – Dr. Sonnhard Graubner Jun 27 '17 at 12:29
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Looks an awful lot like this question. See, for example, Hagen von Eitzen's answer to the duplicate target in how to use convolution here. The trick is to replace the other $\binom n k$ factor with $\binom n {n-k}$. – Jyrki Lahtonen Jun 27 '17 at 12:31
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Also, welcome to Math.Stackexchange. It was a bit of bad luck that your first question happens to have been handled already. – Jyrki Lahtonen Jun 27 '17 at 12:35