# Motzkin numbers and related sequences modulo powers of 2

### (28 pages)

**Abstract.**
We show that the generating function
Σ_{n>=0}*M*_{n}z^{n} for Motzkin numbers *M*_{n}, when
coefficients are reduced modulo a given power of 2, can be
expressed as a polynomial in the basic series
Σ_{e>=0}*z*^{4e}/(1-*z*^{2.4e}) with
coefficients being Laurent polynomials in *z* and 1-*z*.
We use this result to determine *M*_{n} modulo 8 in terms
of the binary digits of *n*, thus improving, respectively
complementing earlier results by Eu, Liu and Yeh
[*Europ. J. Combin.* **29** (2008), 1449-1466]
and by Rowland and Yassawi
[*J. Théorie Nombres Bordeaux* **27** (2015), 245-288].
Analogous results are also shown to hold for related
combinatorial sequences, namely for the Motzkin prefix numbers,
Riordan numbers, central trinomial numbers, and for the sequence of
hex tree numbers.

The following versions are available:

The paper is accompanied by the following *Mathematica* files:
By using the notebook (which requires the other file as input file),
you are able to
redo (most of) the computations that are presented in
this article.

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