bibkey: bala2026a383165 authors: Seiichi Manyama; Peter Bala year: 2026 title: “OEIS A383165, Expansion of e.g.f. log(1 + (exp(2x) - 1)/2)^2 / 2.” doi: null url: https://oeis.org/A383165 claim: “Expansion of e.g.f. log(1 + (exp(2x) - 1)/2)^2 / 2. Conjecture: the sequence {a(n)} reduced modulo a positive integer k is eventually periodic.” strata_touched:

• D5/S1/Recurrence/Periodic/ScaledLogColumnPeriodicity license: citation-only triage: anchor

OEIS A383165

Seiichi Manyama’s entry, dated April 18, 2025, specifies the exponential generating function in the NAME quoted above. Peter Bala’s conjecture, dated February 17, 2026, asserts eventual periodicity modulo every positive integer. The sequence uses offset-zero indexing: a(n) is n! times the coefficient of x^n in that exponential generating function.

The formalization proves the assertion for every natural column r of log(1 + (exp(2x) - 1)/2)^r/r!, and specializes to r = 2 for this entry. With H = (1 + exp(2x))/2 and U = H^(-1), differentiating U^j * (log H)^r/r! gives an integral recurrence. Its factorial-scaled coefficients are integers. Modulo k, the auxiliary index j can be reduced modulo k, and columns at most r form a finite deterministic state. A repeated state gives an eventual period. No particular onset or least period is claimed.

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• URL: https://oeis.org/A383165