bibkey: hanna2025a380060
authors: Paul D. Hanna
year: 2025
title: "OEIS A380060, bilateral unit-quotient square series"
doi: null
url: https://oeis.org/A380060
claim: "Conjecture: for n > 0, a(n) == 2 (mod 4) iff n is a square, and a(n) is divisible by 4 when n is nonsquare."
strata_touched:
  - D5/S1/Recurrence/Invariants/HannaBilateralThetaModFour
license: citation-only
triage: anchor

OEIS A380060

The unpaginated entry’s Name is: “G.f. A(x) satisfies A(x) = Sum_{n=-oo..+oo} x^(n^2) * (A(x) - x^n)^n/(1 - x^n*A(x))^n.”

Its Comments state: “The term x^(n^2) * (A - x^n)^n/(1 - x^n*A)^n is invariant under sign reversal of n.”

Formula (2) is: “A(x) = 1 + 2Sum_{n>=1} x^(n^2) * (A(x) - x^n)^n/(1 - x^nA(x))^n.”

The Comments conjecture: “Conjecture: for n > 0, a(n) == 2 (mod 4) iff n is a square, and a(n) is divisible by 4 when n is nonsquare.”

The formal equation uses formula (2) over rational power series. At degree m, only indices 1 through m are needed because summand n contains X^(n^2). An integer coefficient iteration constructs the solution. Modulo two the solution is one, so the summands become the positive-square monomials. The parity of half of a positive coefficient gives its residue modulo four. The asymptotic formula in the entry is outside this result’s scope.

Verified locator

Source: https://oeis.org/A380060