%I #19 Mar 29 2022 02:33:16
%S 1,1,3,19,145,1401,15331,198283,2840769,45744625,807769891,
%T 15590922051,325339538833,7316871562729,175934564213955,
%U 4508362093795771,122558873094082561,3522465207528093153,106681726559176156099,3395601487535927589235,113287948824653903674641
%N E.g.f.: exp(x/Product_{k>0} (1 - x^k)).
%C From _Peter Bala_, Mar 25 2022: (Start)
%C The sequence terms are odd. 3 divides a(3*n+2), 5 divides a(5*n+4), 9 divides a(9*n+8), 15 divides a(15*n+14) and 19 divides a(19*n+3).
%C More generally, 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, taken modulo 7 the sequence becomes [1, 1, 3, 5, 5, 1, 1, 1, 3, 3, 5, 5, 1, 1, ...], a purely periodic sequence with period 7. (End)
%H Seiichi Manyama, <a href="/A293527/b293527.txt">Table of n, a(n) for n = 0..423</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*A000041(k-1)*a(n-k)/(n-k)! for n > 0.
%t nmax = 25; CoefficientList[Series[E^(x/QPochhammer[x, x]), {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Oct 11 2017 *)
%o (PARI) N=66; x='x+O('x^N); Vec(serlaplace(exp(x/prod(k=1, N, (1-x^k)))))
%Y Cf. A000041, A293528.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Oct 11 2017