bibkey: hanna2025a386666
authors: Paul D. Hanna
year: 2025
title: OEIS A386666, g.f. satisfying A(x)^2 = A(x^2 + 4*A(x)^3)
doi: null
url: https://oeis.org/A386666
claim: "G.f. A(x) satisfies: A(x)^2 = A( x^2 + 4*A(x)^3 ). Conjecture: a(3*n) (mod 3) == 2*a(3*n+1) (mod 3) == a(3*n+2) (mod 3) == 2*A039966(n) for n > 0, where A039966(n) equals the number of partitions of n into distinct powers of 3. A039966(n) is 1 if every base-3 digit of n is 0 or 1, and is 0 otherwise."
strata_touched:
  - D5/S1/Recurrence/Invariants/SignedCatalanCubicSubstitutionModThree
license: citation-only
triage: anchor

OEIS A386666

Paul D. Hanna’s entry specifies the generating function with constant coefficient zero and coefficient of x equal to one. Its conjecture relates three consecutive coefficient residues to A039966. A partition into distinct powers of three is uniquely determined by ternary digits restricted to zero and one; the empty partition gives A039966(0) = 1.

The formalization constructs an integer series by compositional inversion, proves the functional equation and uniqueness, and identifies its reduction modulo three with the signed Catalan series. The digit-support generating function yields the three residue formulas.

Verified locator

• URL: https://oeis.org/A386666
• Auxiliary sequence: https://oeis.org/A039966