%I #15 Mar 12 2021 22:24:48
%S 1,-2,0,1,0,0,1,0,0,0,-2,0,1,-2,0,2,0,0,1,0,0,1,-2,0,2,-4,0,3,-2,0,2,
%T 0,0,1,-4,0,4,-6,0,5,-2,0,3,0,0,3,-6,0,6,-10,0,8,-4,0,5,-2,0,4,-10,0,
%U 9,-14,0,12,-6,0,8,-2,0,7,-14,0,14,-22,0,18,-10
%N Expansion of phi(-x) / psi(-x^3) in powers of x where psi(), 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="/A260162/b260162.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 q^(3/8) * eta(q)^2 * eta(q^6) / (eta(q^2) * eta(q^3) * eta(q^12)) in powers of q.
%F Euler transform of period 12 sequence [ -2, -1, -1, -1, -2, -1, -2, -1, -1, -1, -2, 0, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (768 t)) = 12^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132217.
%F G.f.: 1 / (Product_{k>0} (1 + x^k) * (1 + x^k + x^(2*k)) * (1 + x^(6*k))).
%F 2 * a(n) = A253243(2*n + 3). a(3*n + 2) = 0.
%F Convolution inverse of A132218.
%e G.f. = 1 - 2*x + x^3 + x^6 - 2*x^10 + x^12 - 2*x^13 + 2*x^15 + x^18 + x^21 + ...
%e G.f. = 1/q^3 - 2*q^5 + q^21 + q^45 - 2*q^77 + q^93 - 2*q^101 + 2*q^117 + ...
%t a[ n_] := SeriesCoefficient[ 2^(1/2) x^(3/8) EllipticTheta[ 4, 0, x] / EllipticTheta[ 2, Pi/4, x^(3/2)], {x, 0, n}];
%t a[n_]:= SeriesCoefficient[EllipticTheta[3,0,-q]*QPochhammer[-q^3, q^6]/ QPochhammer[q^6,q^6], {q, 0, n}]; Table[a[n], {n,0,50}] (* _G. C. Greubel_, Mar 19 2018 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^6 + A) / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)), n))};
%o (PARI) q='q+O('q^99); Vec(eta(q)^2*eta(q^6)/(eta(q^2)*eta(q^3)*eta(q^12))) \\ _Altug Alkan_, Mar 20 2018
%Y Cf. A132217, A132218, A253243.
%K sign
%O 0,2
%A _Michael Somos_, Nov 09 2015