TITLE: a Hankel matrix of involution numbers
QUESTION [11 upvotes]: Let $I_k$ denote the enumeration of involutions among permutations in $\mathfrak{S}_k$. I always enjoy these numbers. Of course, here is yet another cute experimental finding for which I ask validity. 

Question. Let $n!!=1!2!\cdots n!$. Is the following true?
  $$\det\left[I_{i+j}\right]_{i,j=0}^n=n!!$$

REPLY [7 votes]: As in arXiv:0902.1650 it suffices to show that $a(n,0)=I_n$ if $a(n,j)$ satisfies $a(n,j)=a(n-1,j-1)+a(n-1,j)+(j+1)a(n-1,j+1)$ with $a(n,-1)=0$ and $a(0,j)=[j=0]$.
But it is easily verified that $a(n,j)=\binom{n}{j}I_{n-j},$ because then the above recursion reduces to $I_n=I_{n-1}+(n-1)I_{n-2}.$
Edit: Another proof along the same lines follows immediately from formula (2.7) in this paper entitled Involutions and their progenies:
$\sum_k\binom{n}{k}I_{n-k} \binom{m}{k}I_{m-k} k!=I_{m+n}.$