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%I #22 Mar 12 2021 22:24:48
%S 1,-3,0,5,1,-3,-7,5,0,0,2,0,1,-3,9,-6,0,0,-7,-11,0,13,9,0,1,10,0,-6,
%T -15,0,-7,0,-15,13,9,0,17,0,0,-11,3,-3,0,5,0,-6,-7,0,17,-19,9,0,-15,0,
%U 0,10,0,-19,0,21,18,10,0,5,0,0,-30,21,-15,-19,-14,0
%N Expansion of f(-x)^3 * psi(x^4) in powers of x where psi(), f() are Ramanujan theta functions.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H Seiichi Manyama, <a href="/A259743/b259743.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^(-5/8) * eta(q)^3 * eta(q^8)^2 / eta(q^4) in powers of q.
%F Euler transform of period 8 sequence [ -3, -3, -3, -2, -3, -3, -3, -4, ...].
%F G.f.: Product_{k>0} (1 - x^k)^3 * (1 + x^(4*k)) * (1 - x^(8*k)).
%e G.f. = 1 - 3*x + 5*x^3 + x^4 - 3*x^5 - 7*x^6 + 5*x^7 + 2*x^10 + x^12 + ...
%e G.f. = q^5 - 3*q^13 + 5*q^29 + q^37 - 3*q^45 - 7*q^53 + 5*q^61 + 2*q^85 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x]^3 EllipticTheta[ 2, 0, x^2] / (2 x^(1/2)) {x, 0, n}];
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x]^3 QPochhammer[ x^8]^2 / QPochhammer[ x^4], {x, 0, n}];
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^8 + A)^2 / eta(x^4 + A), n))};
%K sign
%O 0,2
%A _Michael Somos_, Jul 05 2015
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