OFFSET
0,2
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], Sep 30 2015
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of eta(q^2)^3 * eta(q^3) * eta(q^12) / (eta(q)^3 * eta(q^4)^3 * eta(q^6) ) in powers of q.
Euler transform of period 12 sequence [3, 0, 2, 3, 3, 0, 3, 3, 2, 0, 3, 2, ...].
G.f.: Product_{k>0} (1 - x^(3*k)) * (1 + x^(6*k)) / ( (1 - x^k) * (1 + x^(2*k)) )^3.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (108)^(-1/2) (t/i)^(-1) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A133637.
a(n) ~ exp(2*Pi*sqrt(n/3)) / (2 * 3^(5/4) * n^(5/4)). - Vaclav Kotesovec, Oct 13 2015
EXAMPLE
G.f. = 1 + 3*q + 6*q^2 + 12*q^3 + 24*q^4 + 45*q^5 + 78*q^6 + 132*q^7 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ 2 EllipticTheta[ 2, Pi/4, q^(3/2)] / EllipticTheta[ 2, Pi/4, q^(1/2)]^3 , {q, 0, n}]; (* Michael Somos, Sep 26 2017 *)
nmax=60; CoefficientList[Series[Product[(1-x^(3*k)) * (1+x^(6*k)) / ( (1-x^k)^3 * (1+x^(2*k))^3 ), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 13 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^3 + A) * eta(x^12 + A ) / (eta(x + A)^3 * eta(x^4 + A)^3 * eta(x^6 + A)), n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Sep 07 2007
STATUS
approved