bibkey: hanna2024a372527
authors: Paul D. Hanna
year: 2024
title: "OEIS A372527, square-composition generating function"
doi: null
url: https://oeis.org/A372527
claim: "Conjecture: a(2^k + 1) == 2 (mod 4) for k >= 0, and a(n) == 0 (mod 4) for n > 0 when n is not of the form 2^k + 1."
strata_touched:
  - D5/S1/Recurrence/Invariants/HannaSquareCompositionModFour
license: citation-only
triage: anchor

OEIS A372527

The unpaginated entry’s Name is: “Expansion of g.f. A(x) satisfying A( A(x)^2 ) = xA(x)/(1 - 2A(x)).”

Its Comments state: “Compare to F( F(x)^2 ) = x*F(x)/(1 - F(x)) when F(x) = x/(1-x).”

The conjecture is: “Conjecture: a(2^k + 1) == 2 (mod 4) for k >= 0, and a(n) == 0 (mod 4) for n > 0 when n is not of the form 2^k + 1.”

The offset is 1 and the data begin “1, 2, 6, 24, 110, 544, 2824, 15168, 83566, 469568, 2680496, 15500000, 90596880, …”. Consequently the zero-residue clause is interpreted for n > 1: the first coefficient is 1. Over rational formal power series, the normalization is constant coefficient zero and coefficient one equal to one. The denominator is inverted in the formal power-series ring. The theorem constructs an integral solution, proves uniqueness, and proves both residue clauses for this interpretation. The literal zero-residue clause at n = 1 is false.

Verified locator

Source: https://oeis.org/A372527