AMC 12 2019a Problem 24

For how many integers $n$ between $1$ and $50$, inclusive, is $$\frac{(n^2-1)!}{(n!)^n}$$ an integer? (Recall that $0! = 1$.)

$\textbf{(A) } 31 \textbf{(B) } 32 \textbf{(C) } 33 \textbf{(D) } 34 \textbf{(E) } 35$