bibkey: hanna2025a384270
authors: Paul D. Hanna
year: 2025
title: OEIS A384270, g.f. satisfying A(x) = A(x^4 + 4*x*A(x)^4)/A(x^3 + 3*x*A(x)^3)
doi: null
url: https://oeis.org/A384270
claim: "G.f. satisfies A(x) = A(x^4 + 4*x*A(x)^4) / A(x^3 + 3*x*A(x)^3). Conjecture: a(n) is odd iff n = 2^k for k >= 0."
strata_touched:
  - D5/S1/Recurrence/Invariants/QuarticCubicCatalanQuotientParity
license: citation-only
triage: anchor

OEIS A384270

Paul D. Hanna’s entry of July 10, 2025 defines the generating function by the displayed quotient and conjectures that its odd coefficients occur exactly at powers of two. The normalization is A(0)=0 and a(1)=1.

The denominator has zero constant coefficient, so division by it is not an operation in the integer power-series ring. The Lean module uses the cross-multiplied identity, constructs its normalized integer solution, and proves uniqueness and the conjectured parity at every positive index. Writing A=X*B with B(0)=1 permits cancellation of X^4 and a contracting update involving only integer addition, subtraction, multiplication, and substitution. Thus the construction requires no divisibility assumption.

Verified locator

• URL: https://oeis.org/A384270