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], 2015-2016.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of (psi(x^6) / psi(x) + psi(x^6) / psi(-x)) / 2 in powers of x^2 where psi() is a Ramanujan theta function.
Euler transform of period 48 sequence [1, 2, 3, 2, 2, 0, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 2, 3, 1, 1, 0, 1, 1, 3, 2, 2, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 1, 0, 2, 2, 3, 2, 1, 0, ...].
a(n) ~ exp(sqrt(n)*Pi)/(8*sqrt(6)*n^(3/4)). - Vaclav Kotesovec, Oct 06 2015
EXAMPLE
G.f. = 1 + x + 3*x^2 + 6*x^3 + 11*x^4 + 19*x^5 + 33*x^6 + 53*x^7 + ...
G.f. = q^5 + q^21 + 3*q^37 + 6*q^53 + 11*q^69 + 19*q^85 + 33*q^101 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ x^(-5/8) EllipticTheta[ 2, 0, x^3] / EllipticTheta[ 2, 0, x^(1/2)], {x, 0, 2 n}];
f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; a:= CoefficientList[Series[f[-x, -x^5]*f[x^3, x^5]/f[-x, -x^2]^2, {x, 0, 60}], x]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jul 31 2018 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, n*=2; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^12 + A)^2 / (eta(x^2 + A)^2 * eta(x^6 + A)), n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Oct 06 2015
STATUS
approved