For questions about the study of finite or countable discrete structures, specifically how to count or enumerate elements in a set (perhaps of all possibilities) or any subset. It includes questions on permutations, combinations, bijective proofs, and generating functions.

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The recurrence $a(n,k) = \sum_{0\leqslant j<n} a(n+j,k-1)$

I'm trying to find a closed form expression to the following recurrence relation: \begin{align} &a(0,k) = 0, \quad \forall k\geqslant 1; \\ &a(1,k) = 1, \quad \forall k\geqslant 1; \\ ...