Comment by Gro-Tsen on Efficiently computing $\prod\limits_{i=1}^{n} A_i$

The lower-right entry seems to be Fibonacci for $k=0$, number of involutions for $k=1$, $n!$ for $k=2$, A167449 for $k=3$ (where no formula is given), and unknown in the OEIS for $k=4$. This does not bode well for a general formula!