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I want to know how to caculate $\binom{2n}{0}^3-\binom{2n}{1}^3+\cdots+(-1)^k\binom{2n}{k}^3 \cdots+\binom{2n}{2n}^3$? The sum equals $ (-1)^{n}\binom{3n}{2n}\binom{2n}{n} $, but I donot know how to get this.

thanks

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    In such cases it is better to answer your own question. That way you can get feedback on your answer, and may learn other answers too.2012-04-09

2 Answers 2

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Look for Dixon's Identity. It is among other places discussed in the following Wikipedia article

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See Recurrences for alternating sums of powers of binomial coefficients, which cites Dixon's Summation of a certain series.