bibkey: bala2022a338040 authors: Peter Bala year: 2022 title: “OEIS A338040 (Kotesovec 2020), conjecture: a(n) mod p purely periodic with period p − 1” doi: null url: https://oeis.org/A338040 claim: “E.g.f.: Sum_{j>=0} 4^j * (exp(j*x) - 1)^j. Conjecture: Let p be prime. The sequence obtained by reducing a(n) modulo p for n >= 1 is purely periodic with period p - 1.” strata_touched:

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

OEIS A338040

Vaclav Kotesovec’s entry, dated October 8, 2020, defines the sequence by the exponential generating function quoted above. Peter Bala’s May 31, 2022 comment states the prime-period conjecture. The asserted period need not be minimal.

The entry’s stated finite formula is a(n) = Sum_{k=0..n} 4^k * k^n * k! * Stirling2(n,k). The offset is zero, with initial values 1, 4, 132, 11140, 1763076. The Lean definition uses this finite formula with Mathlib’s Nat.stirlingSecond; it does not separately formalize the exponential generating function identity.

The proof restricts the sum modulo p to k < p. Terms beyond n vanish by the Stirling diagonal bound, and terms with k >= p vanish because p divides k!. Factorial-weighted Stirling inclusion-exclusion then expresses the result as a sum of powers (k*j)^n with weights independent of n, including 4^k. Fermat’s little theorem, together with vanishing of positive powers of zero, gives the conjecture. Induction gives every nonnegative multiple of p - 1.

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• URL: https://oeis.org/A338040