On the g-Ary Expansions of Apéry, Motzkin, Schröder and Other Combinatorial Numbers

Abstract

Let g ≥ 2 be an integer and let (un)n≥1 be a sequence of integers which satisfies a relation un+1 = h(n)un for a rational function h(X). For example, various combinatorial numbers as well as their products satisfy relations of this type. Here, we show that under some mild technical assumptions the number of nonzero digits of un in base g is large on a set of n of asymptotic density 1. We also extend this result to a class of sequences satisfying relations of second order un+2 = h1(n)un+1 + h2(n)un with two nonconstant rational functions \({h_1(X), h_2(X) \in {\mathbb Q} [X]}\). This class includes the Apery, Delannoy, Motzkin, and Schroder numbers.