Dixon's Identity

Theorem
For $n \in \Z_{\ge 0}$:


 * $\displaystyle \sum_{k \mathop \in \Z} \left({-1}\right)^k \binom {2 n} {n + k}^3 = \dfrac {\left({3 n}\right)!} {\left({n!}\right)^3}$

Proof
Follows directly from Dixon's Identity/General Case:


 * $\displaystyle \sum_{k \mathop \in \Z} \left({-1}\right)^k \dbinom {l + m} {l + k} \dbinom {m + n} {m + k} \dbinom {n + l} {n + k} = \dfrac {\left({l + m + n}\right)!} {l! \, m! \, n!}$

setting $l = m = n$.