The first descent in a standard Young tableau of shape $(n,n,n)$

Let $a(n)$ be the number of standard Young tableaux of shape $(n,n,n)$ whose entry in row $2$, column $1$ is odd; equivalently, the number of those whose first descent is even. This is entry A011553 of the On-Line Encyclopedia of Integer Sequences, contributed in 1996, and after thirty years of cura