At first when I read about this problem from a certain Indonesian olympiad, I was puzzled as to how would one solve it. It was a pretty nice revelation of strength of proof methods after I solved it.
Find a closed form for where . The summmation runs over every possible non negative solution of the equation.
First, I observed that a particular product form in the summation like meant selecting one element out of every block created by the composition. This was a clear invitation to a bijection, I just had to find another way of doing it. We can separate with slashes where the alternating slashes could mean selecting an element out of a block and closure of a block. Some more tinkering lead to a neat closed form.