A247343 Moebius transform applied four times to sequence 1,0,0,0,....

1, -4, -4, 6, -4, 16, -4, -4, 6, 16, -4, -24, -4, 16, 16, 1, -4, -24, -4, -24, 16, 16, -4, 16, 6, 16, -4, -24, -4, -64, -4, 0, 16, 16, 16, 36, -4, 16, 16, 16, -4, -64, -4, -24, -24, 16, -4, -4, 6, -24, 16, -24, -4, 16, 16, 16, 16, 16, -4, 96, -4, 16, -24, 0, 16, -64, -4, -24, 16, -64

Offset: 1

Author

Enrique Pérez Herrero, Sep 14 2014

Keywords

• sign
• mult
• easy

Comments

Multiplicative because the Moebius transform of a multiplicative sequence is multiplicative. - Andrew Howroyd, Jul 25 2018

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Enrique Pérez Herrero)

Crossrefs

Cf. A000007, A008683, A007427, A007428.

Programs

Mathematica

tau[1, n_Integer]:=1; SetAttributes[tau, Listable];
tau[k_Integer, n_Integer]:=Plus@@(tau[k-1, Divisors[n]])/; k > 1;
tau[k_Integer, n_Integer]:=Plus@@(tau[k+1, Divisors[n]]*MoebiusMu[n/Divisors[n]]); k<1;
Table[tau[-4, n], {n, 70}]
f[p_, e_] := (-1)^e * Binomial[4, e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 11 2020 *)

PARI

seq(n)={my(v=vector(n, n, n==1)); for(k=1, 4, v=dirmul(v, vector(#v, n, moebius(n)))); v} \\ Andrew Howroyd, Jul 25 2018

PARI

for(n=1, 100, print1(direuler(p=2, n, (1 - X)^4)[n], ", ")) \\ Vaclav Kotesovec, Feb 22 2021

Formula

Dirichlet g.f.: 1/zeta(s)^4.

Multiplicative with a(p^e) = (-1)^e * binomial(4, e). - Amiram Eldar, Sep 11 2020