2-Adic valuations of coefficients of certain integer powers of formal power series

Abstract

Let [Formula: see text] be an integer sequence and [Formula: see text] be its ordinary generating function. In this paper, we study the behavior of 2-adic valuations of the sequence [Formula: see text], where [Formula: see text] is fixed and [Formula: see text] More precisely, we propose a method, which under suitable assumptions on the sequence [Formula: see text] allows us to prove boundedness of the sequence [Formula: see text] for certain values of [Formula: see text]. In particular, if [Formula: see text] is the classical Rudin–Shapiro sequence, then we prove that [Formula: see text] for given [Formula: see text] and all [Formula: see text]. A similar result is proved for a relative of the Rudin–Shapiro sequence recently introduced by Lafrance, Rampersad and Yee.