bibkey: bala2022a220181 authors: Peter Bala year: 2022 title: “OEIS A220181, e.g.f. Sum_{n>=0} (1 - exp(-nx))^n” doi: null url: https://oeis.org/A220181 claim: “E.g.f.: Sum_{n>=0} (1 - exp(-nx))^n. a(n) = Sum_{k=0..n} (-1)^(n-k) * k^n * k! * Stirling2(n,k). 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, 0, 3, 0, 0, 1, 1, 0, 3, 0, 0, 1 …], with an apparent period of 6. Cf. A122399. - Peter Bala, Jun 01 2022” strata_touched:

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

OEIS A220181

The entry gives the exponential generating function and offset-zero finite formula quoted above. Peter Bala’s June 1, 2022 comment conjectures pure periodicity modulo every prime at positive indices, with period p - 1. The conjecture does not assert that this period is minimal.

The formal sequence uses the finite formula. For any integer-valued weight w, the sum of w(k) k^n k! Stirling2(n,k) has period p - 1 modulo p for n >= 1. The A220181 formula equals (-1)^n times that sum with w(k) = (-1)^k. For odd primes the period is even; modulo 2 the sign is trivial. Thus the general theorem proves the quoted A220181 conjecture. The exponential generating function identity is not separately formalized.

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• URL: https://oeis.org/A220181