I know that the sum of squares of binomial coefficients is just ${2n}\choose{n}$ but what is the closed expression for the sum ${n\choose 0}^2 - {n\choose 1}^2 + {n\choose 2}^2 + \cdots + (-1)^n {n\choose n}^2$.
Alternating sum of squares of binomial coefficients
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combinatorics
sequences-and-series
binomial-coefficients
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0Can you do it with generating functions? – 2012-08-08
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0I 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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0Wolfram|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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1I don't really understand why combinatorial proof went more or less unnoticed (while standard application of generating functions is heavily upvoted). – 2013-11-30