OFFSET
0,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
FORMULA
Expansion of q^(-1/3)*eta(q)^3*eta(q^6)^2/(eta(q^2)^2*eta(q^3)) in powers of q.
Euler transform of period 6 sequence [ -3, -1, -2, -1, -3, -2, ...].
a(n) = b(3n+1) where b(n) is multiplicative and b(2^e) = -3(1+(-1)^e)/2 if e>0, b(3^e) = 0^e, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6).
a(4n+3) = 0.
a(n) = (-1)^n * A129576(n). - Amiram Eldar, Jan 28 2024
MATHEMATICA
A122861[n_] := SeriesCoefficient[(QPochhammer[q]^3*QPochhammer[q^6]^2)/(QPochhammer[q^2]^2 *QPochhammer[q^3]), {q, 0, n}]; Table[A122861[n], {n, 0, 50}] (* G. C. Greubel, Oct 05 2017 *)
PROG
(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)^3*eta(x^6+A)^2/eta(x^2+A)^2/eta(x^3+A), n))}
(PARI) {a(n)=local(A, p, e); if(n<0, 0, n=3*n+1; A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, 3*(e%2-1), if(p==3, 0, if(p%6==1, e+1, !(e%2)))))))}
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Sep 15 2006
STATUS
approved