$a(n)$ is the number of linear extensions of poset $P_n$. (Exercise J is an simpler example of counting linear extensions.)
One can figure out what the Hasse diagram of $P_n$ is by observing Hasse diagrams of $P_2$, $P_3$, $P_4$ and $P_5$ in the picture.
Find $a(n)$.
Updated 04/13/2015 and remarked by Ran Pan
$a(n)$ also counts the number of maximum packings of pattern $Q$ in column-strict arrays of size $3\times (n+1)$.
$a(n)$ also counts the number of standard Young tableaux of shape $n^3$ (French notation) avoiding pattern $T$, i.e., the number of standard Young tableaux of shape $n^3$ (French notation) such that for any element in the tableau, its upper element is larger than its right element.
Updated 02/26/2015 and contributed by Quang Tran Bach
$a(n)$ counts sequence A181197 on OEIS.
The hook-length formula for shifted Young tableaux is used for this problem. Solution with details will be posted here soon.