I need to calculate the number of ways of distributing $n$ balls among $k$ boxes, each box may contain no ball, but if it contains any, then it must contain $\geq L$ & $\leq M$ balls.
This effectively solves:
$x_1+x_2+x_3+\dotsb+x_k = n; \quad x_i\in [0,L,L+1,L+2,\dotsc,M-1,M]$.
Is there a known solution to this? Googling "bounded combinatorics" and similar doesn't reveal anything, except for the post below which is a solution for an upper-bound.
It feels like there should be a solution to the $L \leq x_i \leq M$ case, and then the $0$-possible case can then (hopefully) be added to this as a solution to "ways to distribute $n$ balls among $k$ boxes such that at least one box contains no balls"