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E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*j^(k-1)*x^(j*k)/k). - Ilya Gutkovskiy, May 28 2018
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(PARI) N=66; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, (1+k*x^k)^(1/k))))
Cf. A168243.