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%I #8 Apr 29 2019 20:37:02
%S 1,1,3,7,24,49,205,411,1668,5011,20095,40191,241372,482745,1931393,
%T 7725627,38629803,77259607,463562851,927125703,5562774334,22251097753,
%U 89004431205,178008862411,1424071142304,4272213426961,17088854190591,68355416767375,410132502535664,820265005071329
%N a(1) = 1; a(n+1) = Sum_{d|n} tau(d)*a(d), where tau = number of divisors (A000005).
%F G.f.: x * (1 + Sum_{n>=1} tau(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)/(i*j))) = Sum_{n>=1} a(n+1)*x^n/n.
%t a[n_] := a[n] = Sum[DivisorSigma[0, d] a[d], {d, Divisors[n - 1]}]; a[1] = 1; Table[a[n], {n, 1, 30}]
%t a[n_] := a[n] = SeriesCoefficient[x (1 + Sum[DivisorSigma[0, k] a[k] x^k/(1 - x^k), {k, 1, n - 1}]), {x, 0, n}]; Table[a[n], {n, 1, 30}]
%o (PARI) a(n) = if (n==1, 1, sumdiv(n-1, d, numdiv(d)*a(d))); \\ _Michel Marcus_, Apr 29 2019
%Y Cf. A000005, A007557, A060640, A307794, A319133.
%K nonn
%O 1,3
%A _Ilya Gutkovskiy_, Apr 29 2019
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