A matrix generalization of a theorem of Fine

, for m in the range 0 ≤ m ≤ n, that are not divisible by p. We give a matrix product that generalizes Fine’s formula, simultaneously counting binomial coefficients with p-adic valuation α for each α ≥ 0. For each n this information is naturally encoded in a polynomial generating function, and the sequence of these polynomials is p-regular in the sense of Allouche and Shallit. We also give a further generalization to multinomial coefficients.