Consider the following recurrence relation:
$$A(h,0)=1\\ A(h,h)=c^h\\ A(h,r)=A(h-1,r)+(c-1)\cdot A(h-1,r-1).$$
Obviously, this is just a generalization of A008949, where $c=2$. Since I'm pretty sure we're not the first ones dealing with it -- is there some source where it was already solved? I'm really only looking for a source to cite, we already have the solution.
Thanks in advance!