On a Sierpiński Pascal Matrix Construction and Monic Irreducible Polynomials over Binary Finite Fields

This research note presents a weighted matrix construction based on the parity pattern of Pascal's triangle modulo 2, represented by the Sierpiński Pascal matrix. By applying a binary carry characterization, the construction yields the sequence a(n) = (8n − 2n)/3, which coincides with the classical enumeration formula for monic irreducible cubic polynomials over GF(2n).