Odd Entries in Pascal's Trinomial Triangle

TL;DR

This paper investigates the distribution of odd coefficients in Pascal's trinomial triangle, reviewing an algorithm for asymptotic analysis and extending it to more complex cases involving additional terms and higher powers.

Contribution

It introduces a method to evaluate the asymptotics of odd coefficients in generalized trinomial triangles using Lyapunov exponents and extends analysis to more complex polynomial cases.

Findings

The algorithm effectively computes asymptotic behavior of odd coefficients.

Extensions to higher-term polynomials are feasible with the method.

Results provide insights into the distribution of odd entries in generalized Pascal triangles.

Abstract

The nth row of Pascal's trinomial triangle gives coefficients of (1+x+x^2)^n. Let g(n) denote the number of such coefficients that are odd. We review Moshe's algorithm for evaluating asymptotics of g(n) -- this involves computing the Lyapunov exponent for certain 2x2 random matrix products -- and then analyze further examples with more terms and higher powers of x.