A306327 Expansion of Product_{k>=1} 1/(1 - mu(k)*x^k), where mu() is the Möbius function (A008683).

1, 1, 0, -1, 0, 0, 1, 0, 0, -1, 2, 0, 1, -2, 2, 0, 4, -4, 1, -4, 6, -2, 8, -8, 6, -7, 13, -10, 13, -16, 17, -17, 22, -25, 29, -26, 40, -37, 40, -50, 58, -56, 69, -75, 82, -90, 108, -110, 128, -133, 158, -168, 185, -207, 229, -238, 281, -298, 328, -357, 405, -417, 477, -518, 564, -608

Offset: 0

Author

Ilya Gutkovskiy, Feb 07 2019

Keywords

• sign

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A008683, A073576, A117209.

Programs

Mathematica

nmax = 65; CoefficientList[Series[Product[1/(1 - MoebiusMu[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 65; CoefficientList[Series[Exp[Sum[Sum[MoebiusMu[j]^k x^(j k)/k, {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d MoebiusMu[d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 65}]

Formula

G.f.: exp(Sum_{k>=1} Sum_{j>=1} mu(j)^k*x^(j*k)/k).