A083703 Expansion of eta(q)^4/eta(q^4) in powers of q.

1, -4, 2, 8, -4, -8, -8, 16, 6, -12, 8, 8, -8, -24, 0, 16, 12, -16, 10, 24, -8, -16, -24, 16, 8, -28, 8, 32, -16, -8, 0, 32, 6, -32, 16, 16, -12, -40, -24, 16, 24, -16, 16, 40, -8, -40, 0, 32, 24, -36, 10, 16, -24, -24, -32, 48, 0, -32, 24, 24, -16, -40, 0, 48, 12, -16, 16, 56, -16, -32, -48, 16, 30, -64, 8, 40, -24

Offset

0,2

Comments

Euler transform of period 4 sequence [ -4,-4,-4,-3,...].

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Formula

G.f.: Product_{n>0} (1-x^n)^4/(1-x^(4n)).

a(0) = 1, a(n) = -(4/n)*Sum_{k=1..n} A285895(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 29 2017

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d, 4)=0, -3, -4), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Jan 07 2017

Mathematica

CoefficientList[QPochhammer[x]^4/QPochhammer[x^4] + O[x]^80, x] (* Jean-François Alcover, Sep 19 2016 *)

Prog

(PARI) a(n)=if(n<0, 0, X=x+x*O(x^n); polcoeff(eta(X)^4/eta(X^4), n))

Crossrefs

A080965(n)=(-1)^n a(n). a(2n)=0 iff n in A004215 (checked up to n=343).

a(2n)=0 iff A005875(n)=0.

Cf. A274327, A285895.

Sequence in context: A250718 A068504 A080965 * A066104 A143095 A141073

Adjacent sequences: A083700 A083701 A083702 * A083704 A083705 A083706

Keyword

sign

Author

Michael Somos, May 04 2003

Status

approved