Number of ways to put N indistiguishable balls into k indistinguishable boxes of limited size.

Question

I've seen how to put N balls into k distinguishable boxes with no size limit (e.g. the multiplicity of an Einstein solid) where ${{N+k-1} \choose {k}}$ is the multiplicity. However, these boxes are both distinguishable and have no size limit.

I found this post that talks about how to handle the issue of size limit, but not indistinguishability. I also do not quite understand the use of generating functions.

I also found this post which seems similar, but I do not wish to throw away any balls.

Any ideas?