%I #11 Sep 08 2022 08:46:12
%S 1,0,-2,-4,-1,8,6,4,-7,-8,-2,-4,10,-8,-4,0,2,16,-2,16,5,-8,0,-12,-12,
%T -16,-2,12,-9,0,6,8,2,16,12,-20,0,-8,22,0,18,8,-32,0,4,8,-26,-28,-13,
%U -8,0,12,-6,24,2,20,18,0,30,-16,-3,-8,-10,20,0,-16,14,-16
%N Expansion of (phi(-x^3) * f(-x^2))^2 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="/A258065/b258065.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 (f(x, x^2) * f(-x))^2 in powers of x where f() is the Ramanujan general theta function.
%F Expansion of q^(-1/6) * (eta(q^2) * eta(q^3)^2 / eta(q^6))^2 in powers of q.
%F Euler transform of period 6 sequence [ 0, -2, -4, -2, 0, -4, ...].
%F a(n) = A030188(3*n).
%e G.f. = 1 - 2*x^2 - 4*x^3 - x^4 + 8*x^5 + 6*x^6 + 4*x^7 - 7*x^8 - 8*x^9 + ...
%e G.f. = q - 2*q^13 - 4*q^19 - q^25 + 8*q^31 + 6*q^37 + 4*q^43 - 7*q^49 + ...
%t a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, x^3] QPochhammer[ x^2])^2, {x, 0, n}];
%t a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2] QPochhammer[ x^3]^2 / QPochhammer[ x^6])^2, {x, 0, n}];
%o (PARI) {a(n) = if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^3 + A)^2 / eta(x^6 + A))^2, n))};
%o (Magma) A := Basis( ModularForms( Gamma0(72), 2), 409); A[2] - 2*A[14];
%o (Magma) A := Basis( CuspForms( Gamma0(72), 2), 409); A[1];
%K sign
%O 0,3
%A _Michael Somos_, May 18 2015