OFFSET
1,2
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1000
FORMULA
Expansion of a(q^3) * b(q) * c(q^3) / 3 in powers of q where a(), b(), c() are cubic AGM theta functions.
Expansion of ((eta(q^3)^3 / eta(q^9))^3 - (eta(q)^3 / eta(q^3))^3) / 9 in powers of q.
a(3*n) = 0.
Conjecture: multiplicative with a(3^e) = 0, a(p^e) = ((p^2)^(e+1)-1)/(p^2-1) for p == 1 (mod 3), a(p^e) = (1-(-p^2)^(e+1))/(p^2+1) for p == 2 (mod 3). - Andrew Howroyd, Aug 05 2018
EXAMPLE
q - 3*q^2 + 13*q^4 - 24*q^5 + 50*q^7 - 51*q^8 + 72*q^10 - 120*q^11 + ...
MATHEMATICA
eta[q_]:= q^(1/24)*QPochhammer[q]; Rest[CoefficientList[Series[ ((eta[q^3]^3/ eta[q^9])^3 - (eta[q]^3/eta[q^3])^3)/9, {q, 0, 50}], q]] (* G. C. Greubel, Aug 11 2018 *)
PROG
(PARI) {a(n) = local(A); if( n<1, 0, A = x * O(x^n); polcoeff( ((eta(x^3 + A)^3 / eta(x^9 + A))^3 - (eta(x + A)^3 / eta(x^3 + A))^3) / 9, n))}
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Apr 01 2012
STATUS
approved