A307488 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).

1, 1, 4, 14, 59, 257, 1185, 5609, 27259, 134911, 678252, 3452924, 17767047, 92248717, 482710548, 2543031236, 13477141627, 71800541745, 384320284096, 2065782153388, 11146084675905, 60346599617759, 327749929622743, 1785153353416807, 9748766110978057, 53367282644562541

Offset: 0

Author

Ilya Gutkovskiy, Apr 10 2019

Keywords

• nonn

Examples

			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 + ...
		

Crossrefs

Cf. A001615, A008683, A292875, A307487.

Programs

Mathematica

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]
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]
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]

Formula

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).

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