bibkey: hanna2025a383377
authors: Paul D. Hanna
year: 2025
title: "OEIS A383377, g.f. satisfying A(x) = Sum_{n>=0} x^n * abs(1/A(x)^n)"
doi: null
url: https://oeis.org/A383377
claim: "G.f. satisfies A(x) = Sum_{n>=0} x^n * abs(1/A(x)^n), where abs(F(x)) equals the series expansion formed by the unsigned coefficients in F(x). Conjecture: a(n) is even for n > 1."
strata_touched:
  - D5/S1/Recurrence/Invariants/AbsoluteReciprocalGeometricParity
license: citation-only
triage: anchor

OEIS A383377

Paul D. Hanna’s entry of May 15, 2025 defines the generating series by the coefficientwise absolute reciprocal sum and conjectures evenness above index one. The Lean module constructs the unique integer series with constant coefficient one satisfying this equation, then proves the conjecture.

The equation is expressed coefficientwise: at degree N, only summands indexed from zero through N contribute, since every later summand is divisible by a higher power of X. Thus the finite coefficient sum represents the stated formal generating-function equation exactly. Reducing modulo two removes coefficientwise absolute values. Finite geometric sums and the unit inverse identity then show that the reduced generating series is 1 + X.

Verified locator

• URL: https://oeis.org/A383377