Prove $\sum_{i=0}^{2n}{(-1)^i \binom{2n}{i}^3} = (-1)^n \frac{(3n)!}{(n!)^3}$

Question

I've tried to look at it combinatorically but I couldn't find a good model for $\binom{2n}{k}^3$, I've also tried to solve it using induction but the changes in steps are too much to keep track of

Can any one give a hint on any kind of proof for this?