OFFSET
0,9
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q^(-23/24) * eta(q^6) * eta(q^12)^2 / (eta(q^3) * eta(q^4)) in powers of q.
Euler transform of period 12 sequence [ 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, -1, ...].
a(n) ~ exp(Pi*sqrt(n/6)) / (12*sqrt(n)). - Vaclav Kotesovec, Jul 11 2016
EXAMPLE
G.f. = 1 + x^3 + x^4 + x^6 + x^7 + 2*x^8 + 2*x^9 + x^10 + 2*x^11 + ...
G.f. = q^23 + q^95 + q^119 + q^167 + q^191 + 2*q^215 + 2*q^239 + q^263 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^(3/2)] EllipticTheta[ 2, 0, x^3] / (4 x^(9/8) QPochhammer[ x^4]), {x, 0, n}];
nmax = 100; CoefficientList[Series[Product[(1+x^(3*k)) * (1-x^(12*k))^2 / (1-x^(4*k)), {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^6 + A) * eta(x^12 + A)^2 / (eta(x^3 + A) * eta(x^4 + A)), n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Jul 24 2015
STATUS
approved