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$$\zeta(3)={5\over2}\sum_{n=1}^{\infty}{(-1)^{n-1}\over n^3{2n\choose n}}$$ was the starting point for Apéry's proof of the irrationality of $\zeta(3)$. [OK, so it's Number Theory, not combinatorics --- but, look! it has a binomial coefficient in it!]. Here is Alf van der Poorten's report.