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 q^(1/24) * eta(q^4)^3 * eta(q^6)^3 / (eta(q^2)^2 * eta(q^3) * eta(q^12)^2) in powers of q.
Euler transform of period 12 sequence [ 0, 2, 1, -1, 0, 0, 0, -1, 1, 2, 0, -1, ...].
a(n) ~ exp(Pi*sqrt(n/6)) / (4*sqrt(n)). - Vaclav Kotesovec, Jul 11 2016
EXAMPLE
G.f. = 1 + 2*x^2 + x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 2*x^7 + 3*x^8 + 4*x^9 + ...
G.f. = 1/q + 2*q^47 + q^71 + 2*q^95 + 2*q^119 + 3*q^143 + 2*q^167 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x] EllipticTheta[ 2, 0, x^(3/2)] / (4 x^(5/8) QPochhammer[ x^2, x^12] QPochhammer[ x^10, x^12] QPochhammer[ x^12]), {x, 0, n}];
nmax = 50; CoefficientList[Series[Product[(1-x^(4*k))^3 * (1-x^(6*k))^3 / ((1-x^(2*k))^2 * (1-x^(3*k)) * (1-x^(12*k))^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 11 2016 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^3 * eta(x^6 + A)^3 / (eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^12 + A)^2), n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Jul 24 2015
STATUS
approved