%I #10 Mar 12 2021 22:24:48
%S 1,-2,0,-1,4,0,2,-6,0,-4,8,0,7,-14,0,-10,24,0,14,-34,0,-22,48,0,33,
%T -72,0,-45,104,0,62,-142,0,-88,192,0,122,-266,0,-163,364,0,216,-480,0,
%U -290,632,0,386,-840,0,-502,1104,0,650,-1426,0,-846,1832,0,1093
%N Expansion of f(-x, -x) * f(-x^3, -x^15) / f(-x^6, -x^12)^2 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="/A261251/b261251.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 phi(-x) * psi(x^9) / (f(-x^6) * psi(x^3)) in powers of x where phi(), psi(), f() are Ramanujan theta functions.
%F Expansion of q^(-1/2) * eta(q)^2 * eta(q^3) * eta(q^18)^2 / (eta(q^2) * eta(q^6)^3 * eta(q^9)) in powers of q.
%F Euler transform of period 18 sequence [ -2, -1, -3, -1, -2, 1, -2, -1, -2, -1, -2, 1, -2, -1, -3, -1, -2, 0, ...].
%F a(3*n) = A261252(n). a(3*n + 1) = -2 * A217786(n). a(3*n + 2) = 0.
%F Convolution inverse of A261240.
%e G.f. = 1 - 2*x - x^3 + 4*x^4 + 2*x^6 - 6*x^7 - 4*x^9 + 8*x^10 + ...
%e G.f. = q - 2*q^3 - q^7 + 4*q^9 + 2*q^13 - 6*q^15 - 4*q^19 + 8*q^21 + ...
%t a[ n_] := SeriesCoefficient[ x^(-3/4) EllipticTheta[ 4, 0, x] EllipticTheta[ 2, 0, x^(9/2)] / (QPochhammer[ x^6] EllipticTheta[ 2, 0, x^(3/2)]), {x, 0, n}];
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^3 + A) * eta(x^18 + A)^2 / (eta(x^2 + A) * eta(x^6 + A)^3 * eta(x^9 + A)), n))};
%Y Cf. A217786, A261240, A261252.
%K sign
%O 0,2
%A _Michael Somos_, Aug 12 2015