OFFSET
0,3
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of eta(q^3)^2 * eta(q^9)^2 / (eta(q^2)^3 * eta(q^18)) in powers of q.
Euler transform of period 18 sequence [0, 3, -2, 3, 0, 1, 0, 3, -4, 3, 0, 1, 0, 3, -2, 3, 0, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 4/9 g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A182036.
a(n) ~ (-1)^n * exp(2^(3/2)*Pi*sqrt(n)/3) / (2^(3/4) * 3^(5/2) * n^(3/4)). - Vaclav Kotesovec, Mar 21 2018
EXAMPLE
G.f. = 1 + 3*x^2 - 2*x^3 + 9*x^4 - 6*x^5 + 21*x^6 - 18*x^7 + 48*x^8 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^9] QPochhammer[ x^3]^2 / QPochhammer[ x^2]^3, {x, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^2 * eta(x^9 + A)^2 / (eta(x^2 + A)^3 * eta(x^18 + A)), n))};
(PARI) lista(nn) = {q='q+O('q^nn); Vec(eta(q^3)^2*eta(q^9)^2/(eta(q^2)^3*eta(q^18)))} \\ Altug Alkan, Mar 21 2018
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Jan 29 2018
STATUS
approved