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

0,3

Links

Table of n, a(n) for n=0..25.

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)).

Example

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 + ...

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]

Crossrefs

Cf. A001615, A008683, A292875, A307487.

Sequence in context: A296943 A104987 A149492 * A351634 A241706 A111276

Adjacent sequences: A307485 A307486 A307487 * A307489 A307490 A307491

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 10 2019

Status

approved