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arXiv · 2506.13375

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

Jinlong Tang·Guoce Xin
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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.

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Jinlong Tang, Guoce Xin. 2025-06-16. Meeting a Challenge raised by Ekhad and Zeilberger related to Stern's Triangle. https://arxiv.org/abs/2506.13375

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