%I #8 Aug 10 2019 04:26:57

%S 1,2,3,1,-1,-6,-3,2,9,9,-6,-24,-25,16,72,75,-35,-213,-239,78,627,767,

%T -182,-1890,-2477,355,5847,8109,-360,-18195,-26801,-1225,56724,89040,

%U 11431,-177897,-297030,-61857,560310,994427,284075,-1766754,-3338212,-1201932

%N Expansion of Sum_{k>=1} mu(k)*log(1 + x^k/(1 - x^k)^3)/k.

%C Inverse Euler transform of triangular numbers (A000217).

%F -1 + Product_{n>=1} 1/(1 - x^n)^a(n) = g.f. of A000217.

%t nmax = 44; CoefficientList[Series[Sum[MoebiusMu[k] Log[1 + x^k/(1 - x^k)^3]/k, {k, 1, nmax}], {x, 0, nmax}], x] // Rest

%t nmax = 50; s = ConstantArray[0, nmax]; Do[s[[j]] = j^2*(j + 1)/2 - Sum[s[[d]]*(j - d)*(j - d + 1)/2, {d, 1, j - 1}], {j, 1, nmax}]; Table[Sum[MoebiusMu[k/d]*s[[d]], {d, Divisors[k]}]/k, {k, 1, nmax}] (* _Vaclav Kotesovec_, Aug 10 2019 *)

%Y Cf. A000217, A000294, A008683, A308291, A316152.

%K sign

%O 1,2

%A _Ilya Gutkovskiy_, May 18 2019