%I #26 Mar 12 2021 22:24:48
%S 1,0,3,-2,9,-6,21,-18,48,-44,99,-102,204,-216,393,-438,747,-846,1362,
%T -1584,2448,-2872,4275,-5082,7356,-8784,12390,-14894,20592,-24798,
%U 33651,-40644,54336,-65640,86535,-104628,136356,-164736,212388,-256498,327690,-395214
%N Expansion of phi(-x^9) * f(-x^3)^2 / f(-x^2)^3 in powers of x where f(), phi() are Ramanujan theta functions.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A298733/b298733.txt">Table of n, a(n) for n = 0..1000</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of eta(q^3)^2 * eta(q^9)^2 / (eta(q^2)^3 * eta(q^18)) in powers of q.
%F Euler transform of period 18 sequence [0, 3, -2, 3, 0, 1, 0, 3, -4, 3, 0, 1, 0, 3, -2, 3, 0, 0, ...].
%F 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.
%F 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
%e G.f. = 1 + 3*x^2 - 2*x^3 + 9*x^4 - 6*x^5 + 21*x^6 - 18*x^7 + 48*x^8 + ...
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^9] QPochhammer[ x^3]^2 / QPochhammer[ x^2]^3, {x, 0, n}];
%o (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))};
%o (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
%Y Cf. A182036.
%K sign
%O 0,3
%A _Michael Somos_, Jan 29 2018