bibkey: hanna2026a397591
authors: Paul D. Hanna
year: 2026
title: "OEIS A397591: linear-exponent coefficient equations and dyadic neighbors"
doi: null
url: https://oeis.org/A397591
claim: "Conjecture: for n > 3, a(n) is odd iff n is of the form 2^k-1 or 2^k+1 (k > 1)."
strata_touched:
  - D5/S1/Recurrence/LinearRows/LinearExponentDyadicSupport
license: citation-only
triage: anchor

OEIS A397591

The NAME reads: “G.f. A(x) satisfies [x^(n-1)] (1 - A(x))^n / (1 - n*x) = 0 for n > 1.” The FORMULA specifies A(x) = Sum_{n>=1} a(n)*x^n; the offset is 1. This fixes the zero constant term and the coefficient indexing. The formal result is over integer power series, and also proves existence and uniqueness.

The quoted comment is explicitly labeled Conjecture in the retrieved entry. Its author field credits Paul D. Hanna, Jul 09 2026. The only link is the 400-term b-file. The asymptotic formula does not prove parity. The b-file was not opened (ASSUMED-UNVERIFIED); the first 21 DATA terms were compared with an exact-integer recurrence probe, which is not a proof of the conjecture.

A397590 is a triangle, A397902 uses squared exponents, and A320415 is linked by a reciprocal generating function. No theorem is transferred from those entries. The parity proof here is a local derivation from normalized rows, coefficientwise stabilization and the Catalan series modulo two. The OEIS entry is the source of the problem, not a source of that proof.

Verified locator

url: https://oeis.org/A397591

doi: null

Official entry views: https://oeis.org/search?q=id:A397591&fmt=json and https://oeis.org/A397591/internal. No DOI is assigned to the entry.

Proof scope

The literature verdict remains open; no global absence of earlier proofs is claimed.