bibkey: hanna2026a397242 authors: Paul D. Hanna year: 2026 title: “OEIS A397242, o.g.f. A(x) = exp(x + Sum (n^2-1) a(n) x^n / n^2), parity conjecture” doi: null url: https://oeis.org/A397242 claim: “O.g.f. satisfies A(x) = exp( x + Sum_{n>=2} (n^2-1) * a(n)*x^n / n^2 ). Conjecture: a(n) is odd iff n+1 is a power of 2.” strata_touched:

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

OEIS A397242

Paul D. Hanna’s entry, dated June 19, 2026, gives the generating equation and the first parity conjecture quoted above. The sequence has offset zero and begins 1, 1, 2, 15, 244, 6420. The two conjectures modulo three are separate statements and are not asserted here.

The formal reading of the exponential equation is A(0)=1 and A’=L’A, where L has zero constant coefficient, coefficient one at degree one, and coefficient (n^2-1)a(n)/n^2 at degrees n>=2. Multiplication by X gives X A’=M A. The integral normalization a(n)=n d(n) for n>=1 makes M(n)=(n^2-1)d(n) for n>=2, with M(0)=0 and M(1)=1. The Lean module constructs this integer solution, proves the differential identity and its exact rational weights, and proves uniqueness among rational solutions with that coefficient shape. Its public equation uses this formal differential reading, rather than a separate exponential-substitution identity.

Modulo two, d(2m)=d(2m-1) for m>=1. The odd-index coefficients e(m)=d(2m+1) satisfy the Catalan recurrence over ZMod(2). Thus X times their generating series equals the reduction of the imported binary Catalan series from D5/S1/Recurrence/Invariants/CatalanCompositionSquareParity. Its binary support proves that a(n) is odd exactly when n+1 is a power of two, including a(0)=1.

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• URL: https://oeis.org/A397242