Dixon's Identity

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


 * $\ds \sum_{k \mathop \in \Z} \paren {-1}^k \binom {2 n} {n + k}^3 = \dfrac {\paren {3 n}!} {\paren {n!}^3}$

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


 * $\ds \sum_{k \mathop \in \Z} \paren {-1}^k \dbinom {l + m} {l + k} \dbinom {m + n} {m + k} \dbinom {n + l} {n + k} = \dfrac {\paren {l + m + n}!} {l! \, m! \, n!}$

setting $l = m = n$.