bibkey: hanna2024a369083 authors: Paul D. Hanna year: 2024 title: “OEIS A369083, g.f. satisfying A(x) = 1 + x*(5A(x)^2 - A(-x)^2)/4” doi: null url: https://oeis.org/A369083 claim: “Expansion of g.f. A(x) satisfying A(x) = 1 + x(5A(x)^2 - A(-x)^2)/4. Conjecture: a(n) == binomial(4n+3,n) (mod 2) for n >= 0 (cf. A263133).” strata_touched:

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

OEIS A369083

Paul D. Hanna’s entry, dated January 12, 2024, gives the generating function and coefficient conjecture quoted above, with offset zero. The formal reading over the integers is the cleared-denominator equation 4*(A-1) = X*(5*A^2-rescale(-1,A)^2), together with constant coefficient one. After embedding into rational power series this is the displayed OEIS equation.

The module constructs an integer solution by degree contraction and proves its uniqueness. At an odd degree the coefficient of an integer series squared is even: reduction modulo two and the Frobenius square identity show that it has only even-degree support. This makes the coefficient division by two exact. Eliminating the reflected series gives a quartic equation. Modulo two, it says F*(1+X*F)^3=1. Thus H=1+X*F satisfies H^4-H^3=X. Factoring the difference of two such equations gives a factor with constant coefficient one, proving uniqueness among normalized series. Comparison with A380709 and the identity choose(4*n+3,n+1)=3*choose(4*n+3,n) prove the coefficient conjecture.

The formal proof uses the frozen prerequisite D5/S1/Recurrence/Parity/AbsoluteReciprocalCubeParity, statement identifier sha256:a7484b050055466f1a3091c41ffc3856281a6de69fd76b14ba71577ef1c07c75. Its integer generating series G satisfies G=1+X*absSeries(invOfUnit(G,1))^3; its reduction modulo two has coefficient one at zero and choose(4*n-1,n) at every positive index.

Verified locator

• URL: https://oeis.org/A369083