bibkey: hanna2023a365095 authors: Paul D. Hanna year: 2023 title: “OEIS A365095, expansion of a generating function defined by a vanishing diagonal” doi: null url: https://oeis.org/A365095 claim: “Expansion of g.f. A(x) satisfying [x^(n-1)] (1 + (n-1)xA(x)^2)^n / A(x)^n = 0 for n > 1. Conjecture: [x^(n-1)] (1 + (k*n-1)xA(x)^2)^n / A(x)^n is divisible by n^2 for n > 0 and all integer k.” strata_touched:

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

OEIS A365095

Paul D. Hanna’s entry, dated September 3, 2023, defines the generating series by the vanishing diagonal in formula (1) and states the square-divisibility conjecture in formula (2). The normalization is A(0)=1. The parameter k ranges over all integers, including zero and negative values.

The formalization constructs the coefficients over the integers. In degree m, the new coefficient has multiplier -(m+1). Differentiation of an (m+1)-st power shows that its degree-m coefficient is divisible by m+1, so the triangular solve is exact. The coefficient-difference calculation yields stabilization, the defining diagonal equation, and uniqueness among normalized integer series.

Changing n-1 to k*n-1 adds n times an integer series to the base of the n-th power. The power difference is divisible by n^2. Multiplication by A^(-n) and coefficient extraction preserve this divisibility. Here A^(-n) means the n-th power of the formal unit inverse, since the constant coefficient is one.

Verified locator

• URL: https://oeis.org/A365095