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%I #9 Mar 09 2020 19:03:11
%S 1,-2,1,0,0,-2,2,0,2,-2,1,0,2,-6,2,0,3,-6,4,0,4,-8,4,0,7,-14,7,0,6,
%T -16,10,0,11,-20,11,0,14,-32,16,0,17,-38,21,0,22,-46,24,0,32,-66,34,0,
%U 34,-78,44,0,49,-96,50,0,60,-130,66,0,72,-154,84,0,90,-186
%N Expansion of (chi(-x) * chi(x^3))^2 in powers of x where chi() is a Ramanujan theta function.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C Convolution square of A328800.
%C G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is g.f. for A328790.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of q^(1/3) * (eta(q) * eta(q^6)^2)^2 / (eta(q^2) * eta(q^3) * eta(q^12))^2 in powers of q.
%F Euler transform of period 12 sequence [-2, 0, 0, 0, -2, -2, -2, 0, 0, 0, -2, 0, ...].
%F G.f.: Product_{k>=1} (1 - x^(2*k-1))^2 * (1 + x^(6*k-3))^2.
%F a(n) = (-1)^n * A328795(n). a(2*n) = A112206(n).
%F a(4*n) = A328789(n). a(4*n + 1) = -2 * A328798(n). a(4*n + 2) = A328790(n). a(4*n + 3) = 0.
%e G.f. = 1 - 2*x + x^2 - 2*x^5 + 2*x^6 + 2*x^8 - 2*x^9 + x^10 + ...
%e G.f. = q^-1 - 2*q^2 + q^5 - 2*q^14 + 2*q^17 + 2*q^23 - 2*q^26 + ..
%t a[ n_] := SeriesCoefficient[ (QPochhammer[ x, x^2] QPochhammer[ -x^3, x^6])^2, {x, 0, n}];
%o (PARI) {a(n) = my(A); if( n < 0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^6 + A)^2)^2 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A))^2, n))};
%Y Cf. A112206, A328789, A328790, A328795, A328798, A328800.
%K sign
%O 0,2
%A _Michael Somos_, Oct 27 2019
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