Diophantine equation of a factorial type I'm interested in nontrivial solutions of Diophantine equations of the type
$$a^2b^3 =  \frac{c!}{(c-k)!} $$
For various values of k fixed, and of course $a,b,c \in \mathbb{Z^+}$
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.
Edit: I changed n to c to emphasize that I am looking for a,b,c that solve this equation. Thus for k= 1, the equation becomes $a^2b^3 = c$, which clearly has infinity many solutions.
 A: You may already know this, but numbers of the form $a^2b^3$ are called powerful numbers. A closely related question that might provide information on your question is to ask for binomial coefficients that are powerful. A Google search of "powerful number" and "binomial coefficient" brought up the following paper of Granville:
On the scarcity of powerful binomial coefficients
Andrew Granville
https://dms.umontreal.ca/~andrew/PDF/powerful.pdf
He proves that there are only finitely many powerful binomial coefficients, contingent on the abc conjecture.
A: You might be interested in extensions to the Sylvester Schur theorem, which by your constraints shows that c is bigger than k^2 as the set of consecutive integers in the product must have a single multiple of q^2 for some prime q bigger than k. A paper of Saradha and Shorey from 2003, Almost Squares and Factorizations in Consecutive Integers, shows the sparsity of solutions to your equation where k-1 of the numbers on the right hand side multiply to a square.  This may be useful for you in a citation search .
Gerhard "Not Quite Almost Powerful Numbers" Paseman, 2020.07.25.
A: The smallest interesting case of $k=2$ reduces to a family of Pell equations paramaterized by $b$:
$$(2c-1)^2 - b^3(2a)^2 = 1.$$
This gives infinitely many solutions.
For example, for $b=2$, we have a series of solutions indexed by $n$:
$$c_n + a_n\sqrt{8} = \frac{(17+6\sqrt{8})^n+1}2.$$
Numerical values of $c_n$ are listed in OEIS A055792.
