slug: oeis-a224899-sinh-power-prime-period bibkey: bala2022a224899 doi: null url: https://oeis.org/A224899 triage: theorem motivation_gids:

• D5/S1/Recurrence/Periodic/SinhPowerPrimePeriod

The prime-period conjecture for OEIS A224899

Problem

OEIS A224899, NAME:

E.g.f.: Sum_{n>=0} sinh(n*x)^n.

COMMENT (Peter Bala, May 29 2022):

Conjecture: Let p be prime. The sequence obtained by reducing a(n) modulo p for n >= 1 is purely periodic with period p - 1. For example, modulo 7 the sequence becomes [1, 1, 2, 0, 3, 0, 1, 1, 2, 0, 3, 0, …], with an apparent period of 6. Cf. A245322.

The Library note attributes the entry to Paul D. Hanna, April 20, 2013, and the conjecture to Peter Bala, May 29, 2022. The conjecture quantifies over all primes. It is false at p = 2; its substantive content survives for every odd prime, so the failure is confined exactly to p = 2.

Motivation

This is a first-tier OEIS conjecture: entry year 2013, conjecture year 2022. KPI = open problems resolved. The resolution refutes the all-primes statement and proves the accompanying unrestricted odd-prime assertion.

Gap

The supplied search report found no proof: the search seat read the OEIS entry and revision history on 2026-09-09 and searched the identifier on arXiv, MathOverflow, and GitHub. This Stage-B seat had no network access and did not independently repeat those searches. No proof found within that scope does not establish exhaustive absence or publication priority.

Route

The escape content is the coefficient bridge H(n,k) = kⁿ·k!·Σ_{r=k..n} C(n,r)(−k)^{n−r}2^{r−k}S(r,k) for the coefficient expression H(n,k) = n![xⁿ] sinh(kx)^k. It exposes the factorial hidden in the raw exponential expansion, giving k! ∣ H(n,k) and H(n,k) = 0 for k > n. The Lean bridge sums over 0 ≤ r ≤ n; the extra terms with r < k vanish. The analytic e.g.f. identity itself is not formalized: H and a start from the finite coefficient formula.

These two vanishing facts collapse the sum modulo p to the fixed window 0 ≤ k < p (a_window, including n = 0). For n ≥ 1, the k = 0 term vanishes, leaving 1 ≤ k < p. For odd p, a_exponential_sum exposes coefficients independent of n. The frozen D5/S1/Recurrence/Periodic/AlternatingWeightStirlingPrimePeriod.pow_add_pred_prime then gives bala_conjecture_odd. The target generality is I; p−1 is a period, not necessarily the least period.

The same analysis identifies the exception: the exponential-sum step needs 2 to be invertible. At p = 2 the claimed period is 1, but a(1) = 1 and a(2) = 8 have different residues. bala_conjecture_two_false proves ¬ balaConjectureTwo. Thus the entry’s single conjecture is refuted exactly at p = 2, with its odd-prime content proved.

Falsifier

The counterexample index is n = 1 at p = 2: a(1 + (2−1)) = a(2) = 8 is even, whereas a(1) = 1 is odd. The orchestrator’s exact numerical check is supporting evidence only; the formal refutation evaluates these definitions inside the proof.

Evidence

• Lean module: D5/S1/Recurrence/Periodic/SinhPowerPrimePeriod.lean.
• Resolution theorem: bala_conjecture_two_false : ¬ balaConjectureTwo; the single Scribe claim is ResolutionKind.Refuted on this theorem.
• Accompanying positive theorem: bala_conjecture_odd, for every odd prime and every n ≥ 1, precisely delimiting where the conjecture is true.
• Companions: coefficient_bridge, a_window, and a_exponential_sum; definitions: H, a, and balaConjectureTwo.
• The supplied module has no declaration named hanna_conjecture.
• Axioms: std3 (propext, Classical.choice, Quot.sound), as reported by the implementation seat’s #print axioms checks.

Triage

theorem. The all-primes conjecture is refuted at p = 2. The accompanying odd-prime theorem preserves the entry’s substantive content for every other prime. There is one dossier and one refutation claim.

ASSUMED-UNVERIFIED

The verbatim quotes and source attribution were supplied by the orchestrator through Library/Recurrence/bala2022a224899.md. The OEIS revision history was read by the search seat, not by this offline Stage-B seat. The supplied literature scope was the OEIS entry and history plus identifier searches on arXiv, MathOverflow, and GitHub on 2026-09-09; exhaustive literature coverage and first-publication priority are not asserted. Identification of the finite coefficient formula with the analytic e.g.f. is not kernel-checked.