bibkey: hanna2025a389542 authors: Paul D. Hanna year: 2025 title: OEIS A389542, g.f. satisfying A(A(x)^2 - x^2) = 4A(x)^3 doi: null url: https://oeis.org/A389542 claim: “G.f. A(x) satisfies A( A(x)^2 - x^2 ) = 4A(x)^3. Conjecture: for n > 1, a(n) == 2 (mod 4) if n = 2^k + 1 (for some k >= 0) otherwise a(n) is divisible by 4.” strata_touched:

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

OEIS A389542

Paul D. Hanna’s entry, authored November 1, 2025, defines the generating series by A(A(x)^2 - x^2) = 4*A(x)^3, with zero constant coefficient and linear coefficient one. It states the coefficient conjecture quoted above. The Lean module constructs the integer series by compositional reversion, proves its equation and uniqueness, and proves both directions of the mod-four classification for every index greater than one.

The integral inverse has the form R = X*(1 + 2*H), where the zero-constant correction satisfies H = -X - H^2 - 2*X*H(4*X^3). Degree contraction constructs H. Modulo two, H is the Catalan series; the binary support theorem from D5/S1/Recurrence/Invariants/CatalanCompositionSquareParity supplies the powers-of-two support. Compositional inversion transports the doubled correction modulo four to A.

Verified locator

• URL: https://oeis.org/A389542