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Uniform Recurrence in the Motzkin Numbers and Related Sequences mod $p$

Nadav Kohen

Abstract

Many famous integer sequences including the Catalan numbers and the Motzkin numbers can be expressed in the form $ConstantTermOf\left[P(x)^nQ(x)\right]$ for Laurent polynomials $Q$, and symmetric Laurent trinomials $P$. In this paper we characterize the primes for which sequences of this form are uniformly recurrent modulo $p$. For all other primes, we show that $0$ has density $1$. This will be accomplished by showing that the study of these sequences mod $p$ can be reduced to the study of the generalized central trinomial coefficients, which are well-behaved mod $p$.

Uniform Recurrence in the Motzkin Numbers and Related Sequences mod $p$

Abstract

Many famous integer sequences including the Catalan numbers and the Motzkin numbers can be expressed in the form for Laurent polynomials , and symmetric Laurent trinomials . In this paper we characterize the primes for which sequences of this form are uniformly recurrent modulo . For all other primes, we show that has density . This will be accomplished by showing that the study of these sequences mod can be reduced to the study of the generalized central trinomial coefficients, which are well-behaved mod .
Paper Structure (5 sections, 9 theorems, 5 equations, 1 figure)

This paper contains 5 sections, 9 theorems, 5 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and Conventions
  3. Central Trinomial Coefficients
  4. Combinations of Central Trinomial Coefficients
  5. A Family of Applicable Sequences

Key Result

Proposition 1

For any prime $p$, the generalized central trinomial coefficients, $a_n = ct\left[P(x)^n\right]$, satisfy $a_n\equiv \prod a_{n_p[i]}\mod p$.

Figures (1)

  • Figure 1: A demonstration of why $a_{n+i,0} = \sum_{j=-i}^ia_{i,j}\cdot a_{n, j}$ when $\alpha_0 = \alpha_1 = 1$. The small red numbers count the number of contributions of each number in a row to the circled $141$.

Theorems & Definitions (22)

  • Definition 1
  • Proposition 1
  • proof
  • Corollary 2
  • Example 3.1
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Lemma 5
  • ...and 12 more