bibkey: hanna2026a397241 authors: Paul D. Hanna year: 2026 title: “OEIS A397241, g.f. with n [x^n] A^n = (n-1) [x^n] A^(n+1), all terms odd” doi: null url: https://oeis.org/A397241 claim: “G.f. A(x) satisfies n * [x^n] A(x)^n = (n-1) * [x^n] A(x)^(n+1) for n > 1. Conjecture: all terms are odd.” strata_touched:

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

OEIS A397241

Paul D. Hanna’s entry, dated June 19, 2026, specifies the generating equation and conjectures that every term is odd. The normalization is a(0) = a(1) = 1; the integer sequence begins 1, 1, 1, 11, 191, 5279. Here [x^n] denotes the degree-n coefficient of a formal power series. Only the oddness conjecture is addressed.

The module constructs the unique normalized integer solution by coefficient stabilization. For two series with constant coefficient one that agree below degree n, the difference of their degree-n equation residuals equals the difference of their degree-n coefficients. Over ZMod 2, the all-ones series satisfies the same equation: central binomial coefficients are even, and Lucas reduction gives evenness of choose(4m+1, 2m) for m > 0. Uniqueness then identifies the reduced series with the all-ones series, proving the conjecture.

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• URL: https://oeis.org/A397241