bibkey: hanna2025a381362 authors: Paul D. Hanna year: 2025 title: “OEIS A381362, bilateral product generating function” doi: null url: https://oeis.org/A381362 claim: “G.f. A(x) satisfies 1/2 = Sum_{n=-oo..+oo} x^nA(x)^n * (A(x)^n + x)^(2n-1) * (x^n + A(x))^(2*n-1). Conjecture: for n > 0, a(n) == 2 (mod 4).” strata_touched:

• D5/S1/Recurrence/Bilateral/BilateralProductModFour license: citation-only triage: anchor

OEIS A381362

Paul D. Hanna’s entry, dated February 21, 2025, specifies the generating function and coefficient conjecture quoted above. The offset is 0,2 and the normalization is A(0) = 1. This entry is the parameter c = 2 of the bilateral product family with exponents c*n-1.

The formalization interprets the literal integer powers in Laurent series over the rationals. At a positive index k the term contains x^k. At index -k, factoring the negative powers gives x^(c*k^2)*A^(c*k^2)*(1+x*A^k)^(-c*k-1)*(1+x^k*A)^(-c*k-1). The two inverse factors are integral unit inverses. Consequently, only indices in [-N,N] contribute to degree N when c >= 2; every larger window gives the same coefficient. The zero term is (1+x)^(-1)*(1+A)^(-1).

Writing H for the nonzero-index sum gives the equivalent polynomial equation (1+x)*(1+A)*(1-2*H) = 2. Degree contraction constructs the unique normalized integer solution. The equation implies that 1+A is twice an integer series; substituting this divisibility back into the equation gives A = 2/(1+x)-1 (mod 4). Each positive coefficient is therefore 2 modulo 4. The general theorem applies to every natural parameter c >= 2.

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• URL: https://oeis.org/A381362