Computing the 4D Geode

TL;DR

This paper computes a specific high-dimensional coefficient related to the hyper-Catalan numbers, demonstrating a novel calculation of the Geode polynomial's value for large parameters, which was previously an open challenge.

Contribution

The paper presents the first computation of G[1000,1000,1000,1000], solving a longstanding challenge in the enumeration of polygon subdivisions.

Findings

Successfully computed G[1000,1000,1000,1000]

Confirmed the structure of the generating series as a zero of a polynomial

Provides a concrete value for a previously unknown coefficient

Abstract

The closed form for the hyper-Catalan number C[m2,m3,m4,...], which counts the number of subdivisions of a roofed polygon into m2 triangles, m3 quadrilaterals, m4 pentagons, etc., has been known since 1940. In 2025, Wildberger and Rubine showed its generating series S[t2,t3,t4,...] is a zero of the general geometric univariate polynomial. They note the factorization S=(t2 + t3 + t4 + ...)G, where the factor G is called the Geode. Later in 2025, Amderberhan, Kauers and Zeilberger issued a challenge to compute G[1000,1000,1000,1000], the coefficient of $t_2^{1000}t_3^{1000}t_4^{1000}t_5^{1000}$ in G. The reward is a donation to OEIS. We describe the computation, give the value and claim the reward.