A329069 Expansion of Product_{k>=1} 1 / (1 + mu(k)^2 * x^k).

1, -1, 0, -1, 2, -2, 1, -2, 4, -3, 2, -4, 7, -6, 4, -8, 12, -10, 9, -14, 20, -18, 16, -24, 32, -29, 27, -38, 49, -46, 43, -59, 74, -71, 69, -90, 112, -107, 106, -136, 164, -160, 159, -199, 238, -232, 234, -288, 338, -333, 338, -412, 477, -473, 485, -582, 667, -666, 686, -813, 923

Offset: 0

Author

Ilya Gutkovskiy, Nov 04 2019

Keywords

• sign

Comments

Convolution inverse of A087188.

Crossrefs

Cf. A005117, A008683, A048165, A073576, A087188, A292561.

Programs

Mathematica

nmax = 60; CoefficientList[Series[Product[1/(1 + MoebiusMu[k]^2 x^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d) Boole[SquareFreeQ[d]] d, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 60}]

Formula

G.f.: Product_{k>=1} 1 / (1 + x^A005117(k)).