bibkey: hanna2025a389476
authors: Paul D. Hanna
year: 2025
title: "OEIS A389476, g.f. satisfying A(x - A(x)^2/(1 - A(x))^2) = x"
doi: null
url: https://oeis.org/A389476
claim: "G.f. A(x) satisfies: A( x - A(x)^2/(1 - A(x))^2 ) = x. Conjecture: a(n) is odd iff n is a term of A027383, where A027383(2*m) = 3*2^m - 2 and A027383(2*m+1) = 4*2^m - 2 for m >= 0."
strata_touched:
  - D5/S1/Recurrence/Invariants/QuadraticReversionDyadicSupportParity
license: citation-only
triage: anchor

OEIS A389476

Paul D. Hanna’s entry, dated October 5, 2025, specifies the generating equation and conjectures the parity classification quoted above.

The Lean module constructs the unique integer power series with constant coefficient zero satisfying the equation, and proves its linear coefficient is one. Its inner series is U = X - (A * invOfUnit (1 - A) 1)^2. Since 1 - A has constant coefficient one, this is exactly the unit-inverse interpretation of the OEIS quotient. The generating theorem also proves the denominator-cleared identity (1 - A)^2 * (X - U) = A^2 and A.subst U = X.

Over ZMod 2, the series whose coefficient at n is one exactly when n + 2 = 3 * 2^m or n + 2 = 4 * 2^m satisfies P = X + X^2 + X^2 * P^2. Clearing the unit denominator and quadratic uniqueness prove that this series satisfies the reduced reversion equation. Uniqueness of reversion identifies it with the reduction of the integer series, proving Hanna’s conjecture for every positive index. The shifted equalities are equivalent to the OEIS natural-subtraction formulas: the two products are at least three and four, respectively.

Verified locator

• URL: https://oeis.org/A389476