bibkey: hanna2025a378580 authors: Paul D. Hanna year: 2025 title: “OEIS A378580, g.f. A(x) satisfies A(x/A(x)) = theta_3(x)” doi: null url: https://oeis.org/A378580 claim: “G.f. A(x) satisfies: A(x/A(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/Residue/QuotientThetaCompositionModFour license: citation-only triage: anchor

OEIS A378580

Paul D. Hanna’s entry, dated January 8, 2025, specifies the quotient composition equation and coefficient conjecture quoted above. The offset is 0,2, and the series has constant coefficient one. The quotient x/A(x) means X multiplied by the formal unit inverse of A.

The module constructs the unique integer solution by triangular correction. Inversion preserves agreement below a degree, and multiplication by X raises that agreement by one degree. This gives the triangular coefficient comparison for the quotient substitution and stabilization of the correction sequence.

Modulo four, the theta series T has the form 1+2S, hence TT=1. The product composition identity for the same theta series in ThetaSelfCompositionModFour therefore becomes a quotient composition identity. Uniqueness identifies the reduced integer solution with T, proving both the square-index residue two and the nonsquare-index residue zero, equivalently divisibility by four.

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• URL: https://oeis.org/A378580