%I #7 May 22 2019 20:59:01

%S 1,-1,4,-7,6,-4,8,-39,37,-16,12,-94,14,-92,384,-591,18,65,20,-1542,

%T 2552,-1948,24,-3606,3151,-8048,20440,-30590,30,33326,32,-135455,

%U 178512,-130816,94968,-35029,38,-523964,1596560,-1749734,42,2521186,44,-8374494,16364502

%N Expansion of Sum_{k>=1} (-1)^(k+1)*k*x^k/(1 - k*x^k).

%F L.g.f.: -log(Product_{k>=1} (1 - k*x^k)^((-1)^(k+1)/k)) = Sum_{n>=1} a(n)*x^n/n.

%F a(n) = Sum_{d|n} (-1)^(d+1)*d^(n/d).

%F a(n) = n + 1 if n is odd prime.

%t nmax = 45; CoefficientList[Series[Sum[(-1)^(k + 1) k x^k/(1 - k x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

%t nmax = 45; CoefficientList[Series[-Log[Product[(1 - k x^k)^((-1)^(k + 1)/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax] // Rest

%t Table[Sum[(-1)^(d + 1) d^(n/d), {d, Divisors[n]}], {n, 1, 45}]

%Y Cf. A002129, A055225, A076717.

%K sign

%O 1,3

%A _Ilya Gutkovskiy_, May 22 2019