Meeting a Challenge raised by Ekhad and Zeilberger related to Stern's Triangle

TL;DR

This paper develops a novel method combining transfer matrices, algebraic generating functions, and P-recursions to compute a complex combinatorial function related to Stern's triangle, successfully calculating a large number $oldsymbol{ ext{omega}(10000)}$.

Contribution

It introduces an integrated approach to evaluate complex generating functions that are difficult to compute directly, solving an open problem posed by Ekhad and Zeilberger.

Findings

Successfully computed omega(10000), a 6591-digit number

Demonstrated the broad applicability of the method to similar combinatorial problems

Established a new framework combining transfer matrices and algebraic functions

Abstract

This paper resolves an open problem raised by Ekhad and Zeilberger for computing $\omega(10000)$, which is related to Stern's triangle. While $\nu(n)$, defined as the sum of squared coefficients in $\prod_{i=0}^{n-1} (1 + x^{2^i} + x^{2^{i+1}})$, admits a rational generating function, the analogous function $\omega(n)$ for $\prod_{i=0}^{n-1} (1 + x^{2^i+1} + x^{2^{i+1}+1})$ presents substantial computational difficulties due to its complex structure. We develop a method integrating constant term techniques, conditional transfer matrices, algebraic generating functions, and $P$-recursions. Using the conditional transfer matrix method, we represent $\omega(n)$ as the constant term of a bivariate rational function. This framework enables the calculation of $\omega(10000)$, a $6591$-digit number, and illustrates the method's broad applicability to combinatorial generating functions.