%I #24 Mar 12 2021 22:24:48
%S 1,-2,0,2,0,-4,1,6,0,-8,0,12,-1,-18,0,24,0,-32,0,44,0,-58,0,76,1,-100,
%T 0,128,0,-164,0,210,0,-264,0,332,-1,-416,0,516,0,-640,-1,790,0,-968,0,
%U 1184,2,-1444,0,1752,0,-2120,1,2560,0,-3078,0,3692,-2,-4420,0
%N Expansion of f(-x, -x^5)^2 / (f(x^2, x^10) * f(x^6, x^18)) in powers of x where f(, ) is Ramanujan's general theta function.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A283023/b283023.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 chi(-x)^2 * chi(x^3)^2 * chi(-x^12) / chi(x^2) in powers of x where chi() is a Ramanujan theta function.
%F Expansion of phi(-x) * chi(x^3)^2 * chi(-x^12) / phi(-x^4) in powers of x where phi(), chi() are Ramanujan theta functions.
%F Expansion of phi(-x) * phi(x^3) / (phi(-x^4) * psi(-x^6)) ih powers of x where phi(), psi() are Ramanujan theta functions.
%F Expansion of q^(3/4) * eta(q)^2 * eta(q^6)^4 * eta(q^8) / (eta(q^2) * eta(q^3)^2 * eta(q^4)^2 * eta(q^12) * eta(q^24)) in powers of q.
%F Euler transform of period 24 sequence [-2, -1, 0, 1, -2, -3, -2, 0, 0, -1, -2, 0, -2, -1, 0, 0, -2, -3, -2, 1, 0, -1, -2, 0, ...].
%F a(n) = A134178(2*n). a(6*n + 2) = A(6*n + 4) = 0.
%F a(2*n + 1) = -2 * A083365(n). a(4*n + 1) = -2 * A081055(n). a(4*n + 3) = 2 * A081056(n).
%F a(6*n) = A029838(n). a(12*n) = A258741(n). a(12*n + 6) = A259774(n). a(24*n + 12) = - A258939(n).
%e G.f. = 1 - 2*x + 2*x^3 - 4*x^5 + x^6 + 6*x^7 - 8*x^9 + 12*x^11 + ...
%e G.f. = q^-3 - 2*q + 2*q^9 - 4*q^17 + q^21 + 6*q^25 - 8*q^33 + 12*q^41 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2]^2 QPochhammer[ -x^3, x^6]^2 QPochhammer[ x^12, x^24] / QPochhammer[ -x^2, x^4], {x, 0, n}];
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] QPochhammer[ -x^3, x^6]^2 QPochhammer[ x^12, x^24] / EllipticTheta[ 4, 0, x^4], {x, 0, n}];
%t a[ n_] := SeriesCoefficient[ 2^(1/2) x^(3/4) EllipticTheta[ 4, 0, x] EllipticTheta[ 3, 0, x^3] / (EllipticTheta[ 4, 0, x^4] EllipticTheta[ 2, Pi/4, x^3]), {x, 0, n}] // Simplify;
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^6 + A)^4 * eta(x^8 + A) / (eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^4 + A)^2 * eta(x^12 + A) * eta(x^24 + A)), n))};
%o (PARI) lista(nn) = {q='q+O('q^nn); Vec(eta(q)^2*eta(q^6)^4*eta(q^8)/(eta(q^2)*eta(q^3)^2*eta(q^4)^2*eta(q^12)*eta(q^24)))} \\ _Altug Alkan_, Mar 21 2018
%Y Cf. A029838, A081055, A081056, A083365, A134178, A258741, A258939, A259774.
%K sign
%O 0,2
%A _Michael Somos_, Feb 26 2017