Let $I_k$ denote the enumeration of involutions among permutations in $\mathfrak{S}_k$. Here is yet another cute finding for which I ask a:
Question. Is there a direct proof (or interpretation or transformation) of the following equality?
$$\det\left[I_{i+j}\right]_{i,j=0}^n =\det\left[(i+1)^j\right]_{i,j=0}^n.$$
Remark 1. I know how to prove them individually, so that's not what we seek here. The RHS is a specialization of Vandermonde.
Remark 2. One may consider looking at the two MO questions/answers: here and also here.
