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bibkey: hanna2026a397349 authors: Paul D. Hanna year: 2026 title: “OEIS A397349, logarithmic-weight coefficients: residue and parity” doi: null url: https://oeis.org/A397349 claim: “L.g.f. Sum_{n>=1} a(n)x^n/n = log(1+x + Sum_{n>=2} 3n/(3*n^2 - 1) * a(n)*x^n ). Conjecture: a(n) is odd iff n is a power of 2. Conjecture: a(n) = [1,2] repeating (mod 3) for n >= 1.” strata_touched:

  • D5/S1/Recurrence/Residue/LogarithmicWeightCatalanParity
  • D5/S1/Recurrence/Residue/LogarithmicWeightBinaryParity license: citation-only triage: anchor

OEIS A397349

Paul D. Hanna’s NAME and both COMMENT lines are retained verbatim:

%N A397349 L.g.f. Sum_{n>=1} a(n)*x^n/n = log(1+x + Sum_{n>=2} 3*n/(3*n^2 - 1) * a(n)*x^n ).
%C A397349 Conjecture: a(n) is odd iff n is a power of 2.
%C A397349 Conjecture: a(n) = [1,2] repeating (mod 3) for n >= 1.

The claim field quotes the source. Both quoted conjectures are settled here for the frozen constructed sequence, not for the entry’s l.g.f. definition: LogarithmicWeightCatalanParity proves the modulo-three alternation and LogarithmicWeightBinaryParity proves the power-of-two parity clause, discharging the parity_conjecture : Prop the first module left unproved. No module identifies that sequence with the generating function of the NAME, so neither result is a statement about the entry as the source defines it.

The integer construction uses a single well-founded recursion for s, with s 0 = s 1 = 0, a n = if n = 1 then 1 else (3n²−1)*s n, b n = if n ≤ 1 then 1 else 3n*s n, and, for n ≥ 2, s n = ∑_{k=1}^{n−1} a k * b (n−k). Modulo three, the explicit factor in b m for m ≥ 2 kills all summands except k=n−1. The private escape witness s_mod_three gives the boundary term; a_mod_three_step then gives alternation by induction from a 1 = 1.

Erratum recorded in Library/Words/oeis2026triage0911b.md: A397348’s %F(6) is misprinted. Its b_n = n·a_n/(n²−1) gives 22/3 at n = 2, whereas the correct b 2 = 6. The formalization uses only %N and the recurrence above; the conjecture COMMENTS are targets, not premises. The module does not prove a power-series identification or uniqueness theorem.

The quotes are supplied by the orchestrator in oeis-A397349.src. This seat has no network and did not retrieve OEIS or search the literature. The snapshot has no references and only a b-file link. The orchestrator’s finite computations are not rerun here and carry no proof weight; the parity clause is proved symbolically in LogarithmicWeightBinaryParity, not by those computations. No literature-priority claim is made.

Verified locator

  • URL: https://oeis.org/A397349