bibkey: hanna2025a384267 authors: Paul D. Hanna year: 2025 title: “OEIS A384267, a(n) = abs([x^n] 1 + x/A(x)^2), conjecture C.1” doi: null url: https://oeis.org/A384267 claim: “a(n) = abs([x^n] 1 + x/A(x)^2), where A(x) is the g.f. (C.1) a(n) == binomial(3n-1,n)/(3n-1) (mod 2) (cf. A006013).” strata_touched:

• D5/S1/Recurrence/Parity/AbsoluteReciprocalSquareShiftParity license: citation-only triage: anchor

OEIS A384267

Paul D. Hanna’s entry, dated June 19, 2025, defines the integer generating series by coefficientwise absolute value of 1 + x/A(x)^2. Its constant coefficient is one. Conjecture C.1 asserts the quoted congruence for positive indices; the quotient is exact natural-number division.

The formal series uses the unit inverse of its square. Degree contraction constructs the normalized solution and proves uniqueness. Modulo two, the shift U = A + 1 satisfies U = X + U^3. Comparison with the ternary Catalan series, followed by exact binomial division and Lucas reduction, gives C.1.

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• URL: https://oeis.org/A384267