bibkey: hanna2025a378581 authors: Paul D. Hanna year: 2025 title: “OEIS A378581, g.f. satisfying A(xA(x)) = theta_3(x)” doi: null url: https://oeis.org/A378581 claim: “G.f. A(x) satisfies: A(xA(x)) = theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2). Conjecture: for n > 0, a(n) == 2 (mod 4) iff n is square, else a(n) is divisible by 4 if n is nonsquare.” strata_touched:

• D5/S1/Recurrence/Parity/ThetaSelfCompositionModFour license: citation-only triage: anchor

OEIS A378581

Paul D. Hanna’s entry, dated January 8, 2025, gives the generating equation and coefficient conjecture quoted above. The series has constant coefficient one, and a(n) denotes its coefficient at natural degree n.

The theta series has coefficient one at degree zero, two at every positive square degree, and zero elsewhere. The module constructs the unique integral solution of the stated self-composition equation by stabilized coefficient iteration. Over the integers modulo four, multiplication by two annihilates the change caused by substituting X*(1+2*S) for X. Thus the solution has the same coefficients modulo four as the theta series, proving both parts of the conjecture for every positive degree.

Verified locator

• URL: https://oeis.org/A378581