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Enumerative combinatorics, graph theory, order theory, posets, matroids, designs and other discrete structures. It also includes algebraic, analytic and probabilistic combinatorics.

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A binomial sum expression

Does anyone know how to show the following combinatorial equality, $\sum_{i=0}^{n}\left(n-i\right)^{2}\binom{2n}{i}=n\cdot4^{n-1}$? By the way, this is not a homework problem, otherwise one would be …
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