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Putnam 1965 A2

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September 27, 2020
Putnambinomial coefficients

Problem Statement

Show that, for any positive integer nn, ∑r=0[(n−1)/2]{n−2rn(nr)}2=1n(2n−2n−1), \sum_{r=0}^{[(n-1)/2]}\left\{\frac{n-2r}n\binom nr\right\}^2 = \frac 1n\binom{2n-2}{n-1}, where [x][x] means the greatest integer not exceeding xx, and (nr)\textstyle\binom nr is the binomial coefficient "nn choose rr", with the convention (n0)=1\textstyle\binom n0 = 1.
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