bibkey: hanna2025a380709 authors: Paul D. Hanna year: 2025 title: ‘OEIS A380709, g.f. satisfying A(x) = 1 + xabs(1/A(x))^3’ doi: null url: https://oeis.org/A380709 claim: ‘G.f. A(x) satisfies A(x) = 1 + xabs( 1/A(x) )^3. Conjecture: a(n) == binomial(4*n-1, n) (mod 2) for n >= 0 (cf. A263132).’ strata_touched:

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

OEIS A380709

Paul D. Hanna’s entry of February 9, 2025 defines the generating function by the displayed equation and states the binomial parity conjecture. Here abs takes the absolute value of each coefficient of the reciprocal series before cubing. The Lean module constructs the unique integer solution with constant coefficient one and proves the conjecture for every positive index. At index zero, the convention binomial(-1, 0) = 1 is represented separately by a_zero : a 0 = 1.

Verified locator

• URL: https://oeis.org/A380709