%I #9 Apr 29 2019 20:37:09
%S 1,1,4,17,123,739,8888,71105,1066698,13867091,249608380,2995300561,
%T 83868424715,1174157946011,28179790775372,676314978609683,
%U 20965764337966871,377383758083403679,14717966565266619443,294359331305332388861,12363091914824209940661,395618941274374718172273
%N a(1) = 1; a(n+1) = Sum_{d|n} sigma(d)*a(d), where sigma = sum of divisors (A000203).
%F G.f.: x * (1 + Sum_{n>=1} sigma(n)*a(n)*x^n/(1 - x^n)).
%F L.g.f.: -log(Product_{i>=1, j>=1} (1 - x^(i*j))^(a(i*j)/j)) = Sum_{n>=1} a(n+1)*x^n/n.
%t a[n_] := a[n] = Sum[DivisorSigma[1, d] a[d], {d, Divisors[n - 1]}]; a[1] = 1; Table[a[n], {n, 1, 22}]
%t a[n_] := a[n] = SeriesCoefficient[x (1 + Sum[DivisorSigma[1, k] a[k] x^k/(1 - x^k), {k, 1, n - 1}]), {x, 0, n}]; Table[a[n], {n, 1, 22}]
%o (PARI) a(n) = if (n==1, 1, sumdiv(n-1, d, sigma(d)*a(d))); \\ _Michel Marcus_, Apr 29 2019
%Y Cf. A000203, A001001, A307793.
%K nonn
%O 1,3
%A _Ilya Gutkovskiy_, Apr 29 2019