A341835 Dirichlet g.f.: 1 / zeta(s)^9.

1, -9, -9, 36, -9, 81, -9, -84, 36, 81, -9, -324, -9, 81, 81, 126, -9, -324, -9, -324, 81, 81, -9, 756, 36, 81, -84, -324, -9, -729, -9, -126, 81, 81, 81, 1296, -9, 81, 81, 756, -9, -729, -9, -324, -324, 81, -9, -1134, 36, -324, 81, -324, -9, 756, 81, 756, 81, 81

Offset: 1

Author

Ilya Gutkovskiy, Feb 21 2021

Keywords

• sign
• mult

Comments

Dirichlet inverse of A111219.

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000

Crossrefs

Row 9 of A346148.

Cf. A007427, A007428, A008683, A111219, A247343, A341831, A341832, A341833, A341834, A341836.

Programs

Mathematica

a[1] = 1; a[n_] := Times @@ ((-1)^#[[2]] Binomial[9, #[[2]]] &/@ FactorInteger[n]); Table[a[n], {n, 58}]

PARI

for(n=1, 100, print1(direuler(p=2, n, (1 - X)^9)[n], ", ")) \\ Vaclav Kotesovec, Feb 22 2021

Formula

Multiplicative with a(p^e) = (-1)^e * binomial(9, e).

a(1) = 1; a(n) = -Sum_{d|n, d < n} tau_9(n/d) * a(d).