2019 AMC 12A 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 \qquad \textbf{(B) } 32 \qquad \textbf{(C) } 33 \qquad \textbf{(D) } 34 \qquad \textbf{(E) } 35

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.