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%I #12 Apr 05 2024 07:54:05
%S 1,2,4,5,8,8,16,25,52,98,192,345,640,1162,2164,4050,7680,14534,27648,
%T 52479,99956,190554,364544,698525,1341848,2580790,4971616,9587565,
%U 18513920,35790276,69271552,134211600,260297012,505286430,981714296,1908881520,3714580480,7233615306
%N Expansion of Sum_{k>=1} mu(k)*log(1 + x^k/((1 - x^k)*(1 - 2*x^k)))/k.
%C Inverse Euler transform of A000225.
%H Vaclav Kotesovec, <a href="/A308447/b308447.txt">Table of n, a(n) for n = 1..1000</a>
%F -1 + Product_{n>=1} 1/(1 - x^n)^a(n) = g.f. of A000225.
%F a(n) ~ 2^n/n. - _Vaclav Kotesovec_, May 28 2019
%t nmax = 38; CoefficientList[Series[Sum[MoebiusMu[k] Log[1 + x^k/((1 - x^k) (1 - 2 x^k))]/k, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
%t nmax = 50; s = ConstantArray[0, nmax]; Do[s[[j]] = j*(2^ j - 1) - Sum[s[[d]]*(2^(j - d) - 1), {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. A000225, A001037, A008683, A059966, A319918.
%K nonn
%O 1,2
%A _Ilya Gutkovskiy_, May 27 2019
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