`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)`.