I'm trying to solve the following recurrence: $$B_m(n+1,k) = kB_m(n,k)+B_m(n,k-1)-{n\choose m} B_m(n-m,k)$$ where $n,k,m\geq 1$ are integers.
We have boundary conditions \begin{align} B_m(n,n)&=1 \\B_m(n,k)&=0 \qquad n<k \\B_m(n,k)&=0 \qquad mk<n \end{align} and some others.
What I tried: Use exponential generating functions w.r.t $m$ to get rid of the binomial coefficient. Let $A$ be the egf $\{B_m\}_1^\infty$. Doing this gets (I think) $$A'(n+1,k) = kA(n,k) + A(n,k-1) - e^x A(n,k)$$
This gives a 2 variable recurrence with a derviative. My thoughts are to use generating functions again to be left with one variable but I'm not sure how to do that with the derivative here.
Thanks.