bibkey: bala2022a305550 authors: Peter Bala year: 2022 title: “OEIS A305550, Expansion of e.g.f. Product_{k>=1} (1 + (exp(x) - 1)^k)” doi: null url: https://oeis.org/A305550 claim: “Expansion of e.g.f. Product_{k>=1} (1 + (exp(x) - 1)^k). Conjecture: Let k be a positive integer. The sequence obtained by reducing a(n) modulo k is eventually periodic with the period dividing phi(k) = A000010(k).” strata_touched:

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

OEIS A305550

Peter Bala’s comment dated July 8, 2022 states the totient-period conjecture quoted above. The entry has offset zero and records the Stirling transform of A088311. Its defining exponential generating function is the product in the title.

Let Q(k) count partitions of k into distinct positive parts, as represented by Mathlib’s Nat.Partition.distincts. Its ordinary generating series is the formal product of 1 + y^j for positive j. Substituting y = exp(x) - 1 gives the e.g.f. defining A305550. Every coefficient involves only finitely many factors, because the j-th nonconstant term has order at least j.

The module identifies the factorial-scaled coefficients with the weighted Stirling transform T(Q,n). The imported totient-period theorem then proves period phi(m) modulo every positive m from n at least m. Neither the onset nor the period is asserted to be minimal.

Verified locator

• URL: https://oeis.org/A305550
• Locator: NAME (entry by Ilya Gutkovskiy, June 15, 2018) and Peter Bala’s July 8, 2022 COMMENT.