is there a definition of the formula you are trying to solve?

No there isn't. That was part 2 of the problem. The bunch of sums and maps produces the results I require as verified by comparison against the results of exhuastive generation of the computation is summarises. I was looking to reduce the computationally intensive work involved in arriving at those results for higher values of $N & $depth.

Or rather, there wasn't. However, ikegami has divined a generalized formula, which is what I was seeking.