`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)`.