bibkey: hanna2023a365574
authors: Paul D. Hanna
year: 2023
title: "OEIS A365574: generating function and parity conjecture"
doi: null
url: https://oeis.org/A365574
claim: "Conjecture 1: a(k) is odd iff k = 2^n - 2 for n >= 1."
strata_touched:
  - D5/S3/Combinatorics/PowerSeries/HannaLagrangeParity
license: citation-only
triage: anchor

OEIS A365574

Verified locator

Source: https://oeis.org/A365574

The entry is unpaginated. Its NAME states:

Expansion of g.f. A(x) satisfying [x^(n-1)] (1 + (n+1)xA(x))^n / A(x)^n = n*(n+2)^(n-2) for n > 1.

Its COMMENTS state:

Conjecture 1: a(k) is odd iff k = 2^n - 2 for n >= 1.

The rational generating function has constant coefficient one. The proposition asserts existence and that every normalized solution has integral coefficients with the stated parity, including k = 0. The proof constructs an integral compositional inverse of a rational function and determines all inverse coefficients by induction. Conjecture 2 modulo 16 remains open here.

Literature-screen completeness is ASSUMED-UNVERIFIED. The source and the screen recorded in issue #18829 are the literature boundary.