OFFSET
0,3
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..2500
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of eta(q^3)^2 * eta(q^4) / (eta(q^2)^2 * eta(q^6)) in powers of q.
Euler transform of period 12 sequence [0, 2, -2, 1, 0, 1, 0, 1, -2, 2, 0, 0, ...].
a(n) ~ (-1)^n * exp(sqrt(n/2)*Pi) / (2^(9/4) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Oct 06 2015
EXAMPLE
G.f. = 1 + 2*q^2 - 2*q^3 + 4*q^4 - 4*q^5 + 8*q^6 - 8*q^7 + 14*q^8 - 16*q^9 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^3] / EllipticTheta[ 4, 0, q^2], {q, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^2 * eta(x^4 + A) / (eta(x^2 + A)^2 * eta(x^6 + A)), n))};
(PARI) q='q+O('q^99); Vec(eta(q^3)^2*eta(q^4)/(eta(q^2)^2*eta(q^6))) \\ Altug Alkan, Jul 31 2018
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Oct 05 2015
STATUS
approved