Polynomial Diagonals in a Pascal-Type Triangular Array

Abstract

Triangular arrays governed by Pascal-type addition often conceal one-dimensional structure when viewed along rows, columns, or diagonals. The array considered here is indexed by c≥0 and k≥c+1. Its top boundary is prescribed, while each interior entry is obtained by adding the entry immediately to its left and the entry one level above-left. The central question is: for a fixed diagonal offset r=k−c, how does A(c,c+r) depend on c? The answer is polynomial. More precisely, the recurrence performs an iterated binomial summation of the boundary sequence. This yields a closed representation in the rising-binomial basis binom(x+j−1,j). The resulting theorem is most naturally stated for an arbitrary boundary sequence; the particular boundary b_m=2^{m−1}(2^m−1) is then a specialization. This formulation deliberately separates the structural theorem from its arithmetic specialization. It also makes clear which assertions follow purely from the recurrence: polynomiality of the diagonals, their degree, the exact leading term, a finite-difference hierarchy, and a product formula for the diagonal generating functions.