A307487 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).

1, 1, 2, 6, 19, 65, 231, 847, 3187, 12223, 47610, 187836, 749055, 3014453, 12226718, 49931342, 205133243, 847224291, 3515681010, 14650664552, 61286007817, 257256430363, 1083272333869, 4574656128903, 19369837160689, 82214738381631, 349743277470990

Offset: 0

Author

Ilya Gutkovskiy, Apr 10 2019

Keywords

• nonn

Examples

			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 + ...
		

Crossrefs

Cf. A000010, A008683, A050391, A307488.

Programs

Mathematica

terms = 27; CoefficientList[1/x InverseSeries[Series[x/(1 + Sum[EulerPhi[k] x^k, {k, 1, terms}]), {x, 0, terms}], x], x]
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]
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]

Formula

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).

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