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%I #12 Mar 22 2021 14:59:39
%S 1,0,2,-2,2,1,2,-12,11,11,2,-49,2,57,108,-200,2,40,2,-391,780,1013,2,
%T -5423,627,4083,6644,-4453,2,-5043,2,-49680,59172,65519,18028,-251062,
%U 2,262125,531612,-861481,2,-515723,2,-1049929,5180382,4194281,2,-27246019,117651
%N Expansion of Sum_{k>=1} x^k/(1 + k*x^k).
%H Seiichi Manyama, <a href="/A308367/b308367.txt">Table of n, a(n) for n = 1..5000</a>
%F L.g.f.: log(Product_{k>=1} (1 + k*x^k)^(1/k^2)) = Sum_{n>=1} a(n)*x^n/n.
%F a(n) = Sum_{d|n} (-d)^(n/d-1).
%F a(n) = 2 if n is odd prime.
%t nmax = 49; CoefficientList[Series[Sum[x^k /(1 + k x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
%t nmax = 49; CoefficientList[Series[Log[Product[(1 + k x^k)^(1/k^2), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax] // Rest
%t Table[Sum[(-d)^(n/d - 1), {d, Divisors[n]}], {n, 1, 49}]
%o (PARI) a(n) = sumdiv(n, d, (-d)^(n/d-1)); \\ _Michel Marcus_, Mar 22 2021
%Y Cf. A076717, A087909, A300786.
%K sign
%O 1,3
%A _Ilya Gutkovskiy_, May 22 2019