Abstract

Given a sequence x = {x_{n}, n ∈ ℕ} with integer values, or more generally with values in a ring of polynomials with integer coefficients, one can form the generalized binomial coefficients associated with x, \binom{n}{m}_{x} = ∏_{l=1}^{m} \frac{x_{n−l+1}}{x_{l}}. In this note we introduce several sequences that possess the following remarkable feature: the fractions \binom{n}{m}_{x} are in fact polynomials with integer coefficients.