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%I #10 Mar 12 2021 22:24:47
%S 1,0,0,-2,-1,0,0,2,-1,0,0,2,2,0,0,0,-2,0,0,0,-1,0,0,-2,0,0,0,-2,1,0,0,
%T 0,2,0,0,2,0,0,0,0,2,0,0,0,0,0,0,-2,1,0,0,2,-2,0,0,-2,-2,0,0,0,-3,0,0,
%U 2,0,0,0,0,2,0,0,0,-2,0,0,0,2,0,0,2,0,0
%N Expansion of phi(-x^3) * f(-x^4) in powers of x where phi(), f() 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="/A226864/b226864.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^(-1/6) * eta(q^3)^2 * eta(q^4) / eta(q^6) in powers of q.
%F Euler transform of period 12 sequence [ 0, 0, -2, -1, 0, -1, 0, -1, -2, 0, 0, -2, ...].
%F G.f.: (Sum_{k in Z} (-1)^k * x^(3*k^2)) * Product_{k>0} (1 - x^(4*k)).
%F a(n) = (-1)^n * A226862(n). a(4*n + 1) = a(4*n + 2) = 0. a(4*n) = A226289(n).
%e 1 - 2*x^3 - x^4 + 2*x^7 - x^8 + 2*x^11 + 2*x^12 - 2*x^16 - x^20 - 2*x^23 + ...
%e q - 2*q^19 - q^25 + 2*q^43 - q^49 + 2*q^67 + 2*q^73 - 2*q^97 - q^121 + ...
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^3] QPochhammer[ q^4], {q, 0, n}]
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^2 * eta(x^4 + A) / eta(x^6 + A), n))}
%Y Cf. A226289, A226862.
%K sign
%O 0,4
%A _Michael Somos_, Jun 20 2013
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