bibkey: hanna2025a380681
authors: Paul D. Hanna
year: 2025
title: "OEIS A380681–A380686, prime parameters and integral bilateral series"
doi: null
url: https://oeis.org/A380681
claim: "Conjecture: if F(x) satisfies p*x = Sum_{n=-oo..+oo} (-1)^n * x^(p*n) * (F(x) + x^n)^(p*n) with F(0) = 1 for some fixed integer p, then F(x) is an integer series in x iff p is prime. The g.f. A(x) of this sequence is the case when p = 2; other cases include A380682 (p=3), A380683 (p=5), A380684 (p=7), A380685 (p=11), and A380686 (p=13)."
strata_touched:
  - D5/S1/Recurrence/Bilateral/HannaPrimeBilateralIntegrality
license: citation-only
triage: anchor

Verified locator

Source: https://oeis.org/A380681

Prime parameters and integral bilateral series

The unpaginated Comments of OEIS A380681 state:

Conjecture: if F(x) satisfies px = Sum_{n=-oo..+oo} (-1)^n * x^(pn) * (F(x) + x^n)^(p*n) with F(0) = 1 for some fixed integer p, then F(x) is an integer series in x iff p is prime. The g.f. A(x) of this sequence is the case when p = 2; other cases include A380682 (p=3), A380683 (p=5), A380684 (p=7), A380685 (p=11), and A380686 (p=13).

The Name states:

G.f. A(x) satisfies 2x = Sum_{n=-oo..+oo} (-1)^n * x^(2n) * (A(x) + x^n)^(2*n) with A(0) = 1.

The formal statement restricts the parameter to natural p >= 2 and uses rational formal power series. Integer powers are interpreted in rational Laurent series. For each degree N, every symmetric window [-K,K] with K >= N+1 gives the same coefficient. Integrality means that every rational coefficient is an integer. Existence is included for all these parameters. Nonpositive parameters are outside this interpretation. Completeness of the literature screen is ASSUMED-UNVERIFIED.