OFFSET
0,2
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 q^(1/3) * eta(q^2) * eta(q^4)^6 / (eta(q)^2 * eta(q^8)^4) in powers of q.
Euler transform of period 8 sequence [ 2, 1, 2, -5, 2, 1, 2, -1,...].
a(n) ~ exp(Pi*sqrt(n/3)) / (2*sqrt(n)). -Vaclav Kotesovec,Oct 14 2015
EXAMPLE
G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 8*x^4 + 12*x^5 + 16*x^6 + 16*x^7 +...
G.f. = 1/q + 2*q^2 + 4*q^5 + 8*q^8 + 8*q^11 + 12*q^14 + 16*q^17 +...
MATHEMATICA
a[ n_]:= SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ -x^2, x^4]^4, {x, 0, n}];
nmax=60; CoefficientList[Series[Product[(1+x^k) * (1-x^(4*k))^6 / ((1-x^k) * (1-x^(8*k))^4), {k, 1, nmax}], {x, 0, nmax}], x] (*Vaclav Kotesovec,Oct 14 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^4 + A)^6 / (eta(x + A)^2 * eta(x^8 + A)^4), n))};
(PARI) q='q+O('q^99); Vec(eta(q^2)*eta(q^4)^6/(eta(q)^2*eta(q^8)^4)) \\Altug Alkan,Aug 01 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos,Jul 27 2015
STATUS
approved