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bibkey: hanna2026a397356 authors: Paul D. Hanna year: 2026 title: “OEIS A397356, reciprocal square-exponent diagonal generating series” doi: null url: https://oeis.org/A397356 claim: “G.f. A(x) satisfies: [x^n] 1/A(x)^(n^2) = [x^n] 1/A(x)^(n^2-1) for n > 1 with A(0) = A’(0) = 1. Conjecture: for n >= 0, a(n) is odd iff n+1 is a power of 2. Conjecture: for n >= 0, a(n) is not divisible by 3 iff 2*(n+1) is a sum of two powers of 3 (A055235).” strata_touched:

  • D5/S1/Recurrence/Residue/ReciprocalSquareExponentDiagonalParity
  • D5/S1/Recurrence/Residue/ReciprocalSquareExponentDiagonalModThree license: citation-only triage: anchor

OEIS A397356

Paul D. Hanna’s entry gives the generating equation and two conjectures quoted verbatim in the claim above. The entry has two conjectures, and both are now proved in this repository: the parity conjecture in the parity module, and the divisibility-by-three conjecture in the mod-three module.

The module constructs reciprocalSeries from a strict-prefix integer recurrence and defines generatingSeries as its unit inverse, with a n = coeff n generatingSeries. It proves the defining equation, a 0 = a 1 = 1, and uniqueness of the inverse pair. All indices and exponents are natural numbers; subtraction in an exponent is truncated natural subtraction.

Modulo two, reciprocalSeries equals one plus the reduction of catalanSeries. The square-exponent diagonal identity for this binary Catalan series, together with the coefficients of its inverse, proves Odd (a n) if and only if n + 1 = 2 ^ k for some natural number k.

Modulo three the reduced inverse is the sparse series S = 1 - sum_{j >= 1} X^((3^j - 1)/2), which satisfies S = 1 + X + X * S^3 because Frobenius turns the cube into an exponent tripling. A universal diagonal identity for any pair S, Q with S * Q = 1 satisfying that equation, proved by cube extraction and a strong-induction coefficient descent, identifies the reduction of the generating series with S^2. Its support gives 3 not dividing a n exactly when 2 * (n + 1) is a sum of two powers of three, with equal exponents and 3^0 allowed.

Verified locator

  • URL: https://oeis.org/A397356