$a(n)$ is the number of lattice paths from $(0,0)$ to $(4n-3,3n-2)$ not above some special barrier (red in pictures). This problem is similar to Problem 2.
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 $n=k$.
Find $a(n)$.