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I know that the sum of squares of binomial coefficients is just ${2n}\choose{n}$ but what is the closed expression for the sum ${n\choose 0}^2 - {n\choose 1}^2 + {n\choose 2}^2 + \cdots + (-1)^n {n\choose n}^2$.

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    Can you do it with generating functions?2012-08-08
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    I thought I had something clever. I have deleted my post until I have a chance to think on the case when $n$ is even. $n$ being odd still yields 0, unless I am totally mistaken.2012-08-08
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    Wolfram|Alpha gives [this closed form](https://www.wolframalpha.com/input/?i=sum+of+%28-1%29^k+binom+%28n%2Ck%29^2+for+k%3D0..n).2012-08-08
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    I don't really understand why combinatorial proof went more or less unnoticed (while standard application of generating functions is heavily upvoted).2013-11-30

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