Motzkin numbers and related sequences modulo powers of 2

Abstract

We show that the generating function ∑n≥0Mnzn for Motzkin numbers Mn, when coefficients are reduced modulo a given power of 2, can be expressed as a polynomial in the basic series ∑e≥0z4e∕(1−z2⋅4e) with coefficients being Laurent polynomials in z and1−z. We use this result to determine Mn modulo 8 in terms of the binary digits of n, thus improving, respectively complementing earlier results by Eu et al. (2008) and by Rowland and Yassawi (2015). Analogous results are also shown to hold for related combinatorial sequences, namely for the Motzkin prefix numbers, Riordan numbers, central trinomial coefficients, and for the sequence of hex tree numbers.