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G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=1} mu(k)^2*x^k*A(x)^k/(1 - x^k*A(x)^k)^2, where mu() is the Möbius function (A008683).
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%I #5 Apr 10 2019 16:50:03

%S 1,1,4,14,59,257,1185,5609,27259,134911,678252,3452924,17767047,

%T 92248717,482710548,2543031236,13477141627,71800541745,384320284096,

%U 2065782153388,11146084675905,60346599617759,327749929622743,1785153353416807,9748766110978057,53367282644562541

%N G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=1} mu(k)^2*x^k*A(x)^k/(1 - x^k*A(x)^k)^2, where mu() is the Möbius function (A008683).

%F G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=1} psi(k)*x^k*A(x)^k, where psi() is the Dedekind psi function (A001615).

%F G.f.: A(x) = (1/x)*Series_Reversion(x/(1 + Sum_{k>=1} psi(k)*x^k)).

%e G.f.: A(x) = 1 + x + 4*x^2 + 14*x^3 + 59*x^4 + 257*x^5 + 1185*x^6 + 5609*x^7 + 27259*x^8 + 134911*x^9 + 678252*x^10 + ...

%t terms = 26; CoefficientList[1/x InverseSeries[Series[x/(1 + Sum[DirichletConvolve[i, MoebiusMu[i]^2, i, k] x^k, {k, 1, terms}]), {x, 0, terms}], x], x]

%t terms = 26; A[_] = 0; Do[A[x_] = 1 + Sum[MoebiusMu[k]^2 x^k A[x]^k/(1 - x^k A[x]^k)^2, {k, 1, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]

%t terms = 26; A[_] = 0; Do[A[x_] = 1 + Sum[DirichletConvolve[i, MoebiusMu[i]^2, i, k] x^k A[x]^k, {k, 1, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]

%Y Cf. A001615, A008683, A292875, A307487.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Apr 10 2019