$a(n)$ is the number of lattice paths from $(0,0)$ to $(3n-2,3n-2)$ not above some special barrier. Exercise L is a similar but simpler exercise.
In this problem, the barriers (in red) are discontinuous diagonals. The barriers of $n=2$, $n=3$ and $n=4$ are illustrated in the following picture, according to which one can figure out what the barrier looks like for $n=k$.
In the following picture, we randomly draw some feasible paths for $n=2$, $n=3$ and $n=4$.
Find $a(n)$.