%I #18 Mar 29 2022 03:40:42
%S 1,1,5,49,409,4841,66541,1006825,17349809,333948529,6997459861,
%T 159199648961,3918175462345,103227624161689,2901807752857469,
%U 86684932131301561,2738566218754961761,91236821580866560865,3196113263245038385189
%N E.g.f.: exp(Sum_{n>=1} n*A000009(n)*x^n).
%C From _Peter Bala_, Mar 28 2022: (Start)
%C The congruence a(n+k) == a(n) (mod k) holds for all n and k.
%C 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.
%C 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)
%H Seiichi Manyama, <a href="/A293841/b293841.txt">Table of n, a(n) for n = 0..418</a>
%H Peter Bala, <a href="/A047974/a047974_1.pdf">Integer sequences that become periodic on reduction modulo k for all k</a>
%F a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k^2*A000009(k)*a(n-k)/(n-k)! for n > 0.
%t 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 *)
%Y Cf. A000009, A293528, A293840.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Oct 17 2017