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It's half the sum of the same thing from $0$ to $2n$, which in turn is easily related to the varance variance of the number of heads in a sequence of $2n$ tosses of a fair coin.

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It's half the sum of the same thing from $0$ to $2n$, which in turn is easily related to the varance of the number of heads in a sequence of $2n$ tosses of a fair coin.