bibkey: hanna2024a372577 authors: Paul D. Hanna year: 2024 title: “OEIS A372577, expansion of g.f. A(x) satisfying A(x)^2 = A(A(xA(x) + xA(x)^2))” doi: null url: https://oeis.org/A372577 claim: “Expansion of g.f. A(x) satisfying A(x)^2 = A(A( xA(x) + xA(x)^2 )). Conjecture: a(6n - 3) == 3 (mod 4) and a(6n - k) == 1 (mod 4) when k = {0,1,2,4,5} for n >= 1.” strata_touched:

• D5/S1/Recurrence/Residue/DoubleCompositionModFourClassification license: citation-only triage: anchor

OEIS A372577

Paul D. Hanna’s entry, dated May 30, 2024, specifies the double-composition equation and the modulo-four conjecture quoted above. The generating series has zero constant coefficient and coefficient one at degree one. The sequence is indexed by a(n) = [x^n]A(x) for n >= 1. The entry also observes that all terms appear to be odd and compares the equation with the single-composition identity for the Catalan function.

The module writes A = x*U and constructs the constant-one series U by coefficient stabilization of a degree-contracting transformation. It proves the defining double composition and uniqueness with the stated normalization. Over the integers modulo four, unit-denominator cancellation identifies the solution with x/(1-x) + 2*x^3/(1-x^6). These quotients denote formal unit inverses. Thus a(n) has remainder three exactly when n mod 6 = 3, and remainder one at every other positive index. This covers both clauses of the conjecture. Oddness follows from this classification.

Verified locator

• URL: https://oeis.org/A372577