bibkey: bala2018a064618 authors: Peter Bala year: 2018 title: “OEIS A064618, Stirling transform of (n!)^2” doi: null url: https://oeis.org/A064618 claim: “A064618 NAME: Stirling transform of (n!)^2. Conjecture: for fixed k = 1,2,…, the sequence a(n) (mod k) is eventually periodic with the exact period dividing phi(k), where phi(k) is the Euler totient function A000010. For example, modulo 10 the sequence becomes (1, 1, 5, 9, 1, 1, 5, 9, …), with an apparent period 1, 1, 5, 9 of length 4 = phi(10) beginning at a(0). - Peter Bala, Jan 15 2018” strata_touched:
- D5/S1/Recurrence/Periodic/StirlingTransformTotientPeriod license: citation-only triage: anchor
OEIS A064618
The A064618 comment is attributed to Peter Bala, January 15, 2018, and is the
conjecture quoted above; the sibling entry A004123 carries its own note,
bala2022a004123. Both finite formulas use the Stirling numbers of the second
kind S(n,k):
- A064618, offset zero:
a(n) = Sum_{k=0}^n (k!)^2 * S(n,k). - A004123, offset one:
a(n) = Sum_{k=0}^{n-1} 2^k * k! * S(n-1,k)forn >= 1.
The module defines T(w,n) = Sum_{k=0}^n w(k) * k! * S(n,k) over the integers.
It proves T(w,n+phi(m)) = T(w,n) modulo every positive m when n >= m,
for every integer weight w. Terms with k >= m vanish modulo m;
inclusion-exclusion gives a fixed sum of powers. Euler’s theorem for unit
bases and vanishing powers for non-unit bases give the period at each prime
power dividing m, and prime-power divisibility combines the congruences.
The weights w(k)=k! and w(k)=2^k give the two assertions. In the entries’
own indexing, the proved onsets are respectively n >= m and n >= m+1.
Neither onset nor period is asserted to be minimal. The broader A004123
conjecture about every e.g.f. G(exp(x)-1) is outside these two assertions.
Verified locator
- A064618 NAME, OFFSET, FORMULA, and COMMENT: https://oeis.org/A064618
- A004123 NAME, OFFSET, FORMULA, and COMMENT: https://oeis.org/A004123