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%I #5 Mar 12 2018 19:22:21
%S 1,3,10,7,26,24,50,-33,163,38,122,-188,170,108,1580,-1793,290,-273,
%T 362,-1678,9404,3248,530,-49092,16251,14862,66340,14000,842,-135556,
%U 962,-429057,547172,258386,509500,-1392821,1370,1043160,4813052,-8088838,1682,-9267612,1850,8218844,53396438
%N L.g.f.: log(Product_{k>=1} (1 + k*x^k)) = Sum_{n>=1} a(n)*x^n/n.
%F L.g.f.: Sum_{j>=1} Sum_{k>=1} (-1)^(j+1)*k^j*x^(j*k)/j = Sum_{n>=1} a(n)*x^n/n.
%F G.f.: Sum_{k>=1} k^2*x^k/(1 + k*x^k).
%F a(n) = Sum_{d|n} (-d)^(n/d+1).
%e L.g.f.: L(x) = x + 3*x^2/2 + 10*x^3/3 + 7*x^4/4 + 26*x^5/5 + 24*x^6/6 + 50*x^7/7 - 33*x^8/8 + 163*x^9/9 + 38*x^10/10 + ...
%e exp(L(x)) = 1 + x + 2*x^2 + 5*x^3 + 7*x^4 + 15*x^5 + 25*x^6 + 43*x^7 + 64*x^8 + 120*x^9 + 186*x^10 + ... + A022629(n)*x^n + ...
%t nmax = 45; Rest[CoefficientList[Series[Log[Product[(1 + k x^k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
%t nmax = 45; Rest[CoefficientList[Series[Sum[Sum[(-1)^(j + 1) k^j x^(j k)/j, {k, 1, nmax}], {j, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]]
%t nmax = 45; Rest[CoefficientList[Series[Sum[k^2 x^k/(1 + k x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
%t a[n_] := Sum[(-d)^(n/d + 1), {d, Divisors[n]}]; Table[a[n], {n, 1, 45}]
%Y Cf. A022629, A076717, A078308.
%K sign
%O 1,2
%A _Ilya Gutkovskiy_, Mar 12 2018
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