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%I #6 Apr 10 2019 16:49:52
%S 1,1,2,6,19,65,231,847,3187,12223,47610,187836,749055,3014453,
%T 12226718,49931342,205133243,847224291,3515681010,14650664552,
%U 61286007817,257256430363,1083272333869,4574656128903,19369837160689,82214738381631,349743277470990
%N G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=1} mu(k)*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} phi(k)*x^k*A(x)^k, where phi() is the Euler totient function (A000010).
%F G.f.: A(x) = (1/x)*Series_Reversion(x/(1 + Sum_{k>=1} phi(k)*x^k)).
%e G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 19*x^4 + 65*x^5 + 231*x^6 + 847*x^7 + 3187*x^8 + 12223*x^9 + 47610*x^10 + ...
%t terms = 27; CoefficientList[1/x InverseSeries[Series[x/(1 + Sum[EulerPhi[k] x^k, {k, 1, terms}]), {x, 0, terms}], x], x]
%t terms = 27; A[_] = 0; Do[A[x_] = 1 + Sum[MoebiusMu[k] 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 = 27; A[_] = 0; Do[A[x_] = 1 + Sum[EulerPhi[k] x^k A[x]^k, {k, 1, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]
%Y Cf. A000010, A008683, A050391, A307488.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Apr 10 2019
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