A080965 Expansion of eta(q^2)^12/(eta(q)^4eta(q^4)^5) 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

Offset: 0

Author

Michael Somos, Feb 28 2003

Keywords

• sign

Comments

Euler transform of period 4 sequence [4,-8,4,-3,...].

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000

Crossrefs

a(n)=A080964(4n)=2*A072071(4n)-A072070(4n).

A083703(n)=(-1)^n a(n). a(2n)=0 iff n in A004215 (checked up to n=343).

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

Programs

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add([-3, 4, -8, 4]
      [1+irem(d, 4)]*d, d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 05 2015

Mathematica

a[n_] := a[n] = If[n==0, 1, Sum[DivisorSum[j, {-3, 4, -8, 4}[[1 + Mod[#, 4]]]*#&]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 25 2015, after Alois P. Heinz *)

PARI

a(n)=local(X); if(n<0,0,X=x+x*O(x^n); polcoeff(eta(X)^-4*eta(X^2)^12*eta(X^4)^-5,n))

Formula

G.f.: Product_{n>0} (1-x^(2n))^12/((1-x^n)^4(1-x^(4n))^5).