OFFSET
1,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
FORMULA
Expansion of (eta(q)*eta(q^3)*eta(q^4)*eta(q^12))^2/(eta(q^2)*eta(q^6))^2 in powers of q.
Euler transform of period 12 sequence [ -2, 0, -4, -2, -2, 0, -2, -2, -4, 0, -2, -4, ...].
Multiplicative with a(2^e) = -(2^e) if e>0, a(3^e) = 1, a(p^e) = (p^(e+1)-1)/(p-1) if p>3.
a(3*n)=a(n), a(4n+2)=-2*a(2*n+1).
a(n) = (-1)^(n+1)*A111932(n).
Dirichlet g.f.: (1 - 5/2^s + 1/2^(2*s-2) ) * (1 - 1/3^(s-1)) * zeta(s-1) * zeta(s). - Amiram Eldar, Sep 12 2023
MATHEMATICA
eta[q_]:= q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[(eta[q] *eta[q^3]*eta[q^4]*eta[q^12])^2/(eta[q^2]*eta[q^6])^2, {q, 0, n}]; Table[a[n], {n, 1, 50}] (* G. C. Greubel, Mar 07 2018 *)
PROG
(PARI) {a(n)=if(n<1, 0, -(-1)^n*sumdiv(n, d, (n/d%2)*d*(d%3>0)))}
CROSSREFS
KEYWORD
sign,easy,mult
AUTHOR
Michael Somos, Jul 30 2006
STATUS
approved