A Multiplicative Combinatorial Array Of A Triangle.

We introduce a triangular array of integers defined by a(n, k) = k(k + 1)n−kfor n ≥ 1 and 1 ≤ k ≤ n, which we call the researcher’s triangle. The construction is multiplicative and differs from classical additive triangles such as Pascal’s. We give combinatorial interpretationsusing a leaders-followers model and a rooted hierarchical structure model. We identify several diagonal sequences as classical figurate numbers, including powers of two (A000079), naturalnumbers (A000027), and oblong numbers (A002378), and establish connections to triangular and tetrahedral numbers. We derive a bivariate generating function and a recurrence relation, and we discuss the row sums.