OFFSET
-1,2
LINKS
G. C. Greubel, Table of n, a(n) for n = -1..1000
FORMULA
Expansion of (eta(q^3)^8 + 4 * eta(q^6)^8) / (eta(q) * eta(q^2) * eta(q^3)^2 * eta(q^6)^2 * eta(q^9) * eta(q^18)) in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = f(t) where q = exp(2 Pi i t).
a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (2^(3/4) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2017
EXAMPLE
1/q + 5 + 7*q + 10*q^2 + 27*q^3 + 38*q^4 + 82*q^5 + 108*q^6 + 207*q^7 + ...
MATHEMATICA
QP = QPochhammer; s = (QP[q^3]^8+4*q*QP[q^6]^8)/(QP[q]*QP[q^2]*QP[q^3]^2* QP[q^6]^2*QP[q^9]*QP[q^18]) + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 16 2015, adapted from PARI *)
eta[q_] := q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[ q*(eta[q^3]^8 + 4*eta[q^6]^8)/(eta[q]*eta[q^2]*eta[q^3]^2*eta[q^6]^2* eta[q^9]*eta[q^18]), {q, 0, 100}], q]; Table[a[[n]], {n, 1, 80}] (* G. C. Greubel, Jul 03 2018 *)
PROG
(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^3 + A)^8 + 4 * x * eta(x^6 + A)^8) / (eta(x + A) * eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^6 + A)^2 * eta(x^9 + A) * eta(x^18 + A)), n))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Aug 19 2012
STATUS
approved