# How to solve multidimensional recurrence relation with a weight coefficient?

I am trying to solve a multidimensional recurrence relation of the form

$$r_{i,j,k} = w_{j,k}r_{i - 1,j - 1,k - 1} + r_{i,j-1,k} + r_{i,j,k-1} - r_{i,j-1,k-1}$$

My goal is to find a closed-form for $r_{i,j,k}$ in terms of $i, j, k$ and $w_{m,n}$ for arbitrary $m, n$ values.

I got really close with the neat technique provided here that used generating functions, but the weight coefficient $w_{j,k}$ is messing things up since it won't let me factor it out of the generating function. Is it possible to solve this equation when $w_{j,k}$ isn't a constant?