bibkey: hanna2024a373308
authors: Paul D. Hanna
year: 2024
title: OEIS A373308, expansion of Product_{n>=0} (1 - x^(2^n))^3
doi: null
url: https://oeis.org/A373308
claim: "Expansion of Product_{n>=0} (1 - x^(2^n))^3. Conjecture: a(3*n+1) and a(3*n+2) are divisible by 3 for n >= 0."
strata_touched:
  - D5/S1/Recurrence/Invariants/CubedBinaryProductModThree
license: citation-only
triage: anchor

OEIS A373308

Paul D. Hanna’s entry of June 20, 2024 defines the sequence by the expansion of Product_{n>=0} (1 - x^(2^n))^3 and states the quoted divisibility conjecture. The infinite product is represented by its exact defining functional equation A(x) = (1 - x)^3 A(x^2) with constant coefficient one: removing the first factor and shifting the remaining factors gives this equation.

The Lean module constructs the integer coefficients recursively, proves this functional equation and uniqueness, and proves the conjecture for every natural index. It uses this characterization rather than a formal infinite-product operator. Reduction modulo three gives (1 - X)^3 = 1 - X^3; strong induction then shows that the reduced series is supported on multiples of three. The other conjectures in the entry are not asserted here.

Verified locator

• URL: https://oeis.org/A373308