A293841 E.g.f.: exp(Sum_{n>=1} n*A000009(n)*x^n).

1, 1, 5, 49, 409, 4841, 66541, 1006825, 17349809, 333948529, 6997459861, 159199648961, 3918175462345, 103227624161689, 2901807752857469, 86684932131301561, 2738566218754961761, 91236821580866560865, 3196113263245038385189

Offset: 0

Author

Seiichi Manyama, Oct 17 2017

Keywords

• nonn

Comments

From Peter Bala, Mar 28 2022: (Start)

The congruence a(n+k) == a(n) (mod k) holds for all n and k.

It follows that the sequence obtained by taking a(n) modulo a fixed positive integer k is periodic with exact period dividing k. For example, the sequence taken modulo 10 becomes [1, 1, 5, 9, 9, 1, 1, 5, 9, 9, ...], a purely periodic sequence with exact period 5.

5 divides a(5*n+2), 7 divides a(7*n+3); 17 divides a(17*n+7), a(17*n+8) and a(17*n+11). (End)

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..418
• Peter Bala, Integer sequences that become periodic on reduction modulo k for all k

Crossrefs

Cf. A000009, A293528, A293840.

Programs

Mathematica

nmax = 20; CoefficientList[Series[E^Sum[k*PartitionsQ[k]*x^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 18 2017 *)

Formula

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k^2*A000009(k)*a(n-k)/(n-k)! for n > 0.