A307778 a(1) = 1; a(n+1) = Sum_{d|n} (-1)^(d+1)*a(d).

1, 1, 0, 1, -1, 0, 0, 1, -2, -1, 0, 1, -2, -1, 1, 1, -3, -2, 0, 1, -2, -1, 1, 2, -5, -5, 3, 2, -2, -1, 2, 3, -6, -5, 2, 2, -4, -3, 3, 2, -5, -4, 3, 4, -4, -5, 6, 7, -13, -12, 7, 5, -3, -2, 5, 5, -8, -7, 5, 6, -7, -6, 8, 5, -11, -13, 8, 9, -8, -6, 9, 10, -17, -16, 12, 8, -6

Offset: 1

Author

Ilya Gutkovskiy, Apr 28 2019

Keywords

• sign

Crossrefs

Cf. A003238, A281487, A307776, A307777, A307779.

Programs

Mathematica

a[n_] := a[n] = Sum[(-1)^(d + 1) a[d], {d, Divisors[n - 1]}]; a[1] = 1; Table[a[n], {n, 1, 77}]
a[n_] := a[n] = SeriesCoefficient[x (1 + Sum[(-1)^(k + 1) a[k] x^k/(1 - x^k), {k, 1, n - 1}]), {x, 0, n}]; Table[a[n], {n, 1, 77}]

Formula

G.f.: x * (1 + Sum_{n>=1} (-1)^(n+1)*a(n)*x^n/(1 - x^n)).

L.g.f.: log(Product_{n>=1} (1 - x^n)^((-1)^n*a(n)/n)) = Sum_{n>=1} a(n+1)*x^n/n.