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
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
Look for Dixon's Identity. It is among other places discussed in the following Wikipedia article
See Recurrences for alternating sums of powers of binomial coefficients, which cites Dixon's Summation of a certain series.