bibkey: hanna2016a271960
authors: Paul D. Hanna
year: 2016
title: "OEIS A271960: a squared compositional generating function and dyadic vanishing"
doi: null
url: https://oeis.org/A271960
claim: "Conjecture: [x^(2^n-2^k)] (x/A(x))^(2^n) = 0 for k = 0..n-2 and n >= 2."
strata_touched:
  - D5/S3/Combinatorics/PowerSeries/HannaA271960DyadicVanishing
license: citation-only
triage: anchor

OEIS A271960

Verified locator

Source: https://oeis.org/A271960

The entry is unpaginated. Its NAME is:

G.f. A(x) satisfies: A(x)^2 = A( (x + 2*A(x)^2)^2 ).

Its OFFSET is 1. Its EXAMPLE begins A(x) = x + 2x^2 + 9x^3 + 50x^4 + 312x^5 + …. Its COMMENTS state:

Conjecture: [x^(2^n-2^k)] (x/A(x))^(2^n) = 0 for k = 0..n-2 and n >= 2. - Paul D. Hanna, Jan 16 2026

The formal statement uses rational power series with zero constant coefficient and linear coefficient one. The quotient x/A(x) is the multiplicative inverse of the series whose nth coefficient is the (n+1)st coefficient of A. It asserts existence and the coefficient identity for every normalized solution.

The proof uses the compositional inverse B and the odd series H=B+2X² with H²=B(X²). Coefficient decimation gives the support restriction, and Lagrange inversion transfers it to the conjectured coefficients.

Literature-screen completeness is ASSUMED-UNVERIFIED; the recorded search found no earlier settlement of this support conjecture.