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E.g.f.: exp(x/Product_{k>0} (1 - x^k)).
3

%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