bibkey: hanna2024a379202
authors: P. D. Hanna
year: 2024
title: OEIS A379202, a bilateral Laurent generating equation
doi: null
url: https://oeis.org/A379202
claim: "Conjecture: a(n) is even for n > 1."
strata_touched:
  - D5/S1/Recurrence/Invariants/HannaBilateralReversionParity
license: citation-only
triage: anchor

OEIS A379202

Verified locator

DOI: null

Source: https://oeis.org/A379202

The Name section states: “G.f. A(x) satisfies 1/x = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n + 2)^(n+1).”

The Comments section states: “Conjecture: a(n) is even for n > 1.” It also states: “It appears that a(n) == 2 (mod 4) at n = A028309(k) for k >= 4.” OEIS entries have named sections rather than page numbers. The offset is 1, and the initial coefficients are 1, 4, 20, 122, 850, 6432, 51324, 424694.

The formalization uses rational Laurent series for the integer powers and inverse. For a power series A with constant coefficient zero and linear coefficient one, the coefficient of degree j-1 in the bilateral sum is the sum over integer indices from -j-2 through j+2. The omitted summands have larger order. Pairing k with -k-2 proves cancellation modulo two after isolating the index -1. Integral compositional reversion then proves the quoted parity conjecture. The separate mod-four observation remains open in this development.

The literature screen is recorded in https://github.com/the-omega-institute/trureturing/issues/18876. Its completeness is ASSUMED-UNVERIFIED.