I'm interested in nontrivial solutions of Diophantine equations of the type $$a^2b^3 = \frac{n!}{(n-k)!} $$

For various values of k fixed, and of course $a,b,n \in \mathbb{Z^+}$

Does anyone have any insight into this type of equation or a good reference for further reading? My search is being swamped by irrelevant results.