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%I #15 Mar 12 2021 22:24:46
%S 1,-2,1,-2,0,4,1,2,-5,0,-5,4,1,-2,-5,0,7,4,7,0,-4,-10,7,-8,0,4,0,-8,2,
%T 0,1,-2,0,2,0,14,7,0,-5,10,-11,-8,-10,-2,0,10,-4,4,0,0,-5,-8,-11,10,0,
%U 0,14,-2,20,0,-11,4,13,2,-5,-14,0,-14,13,0,-11,-14,8,-2,0,10,13,-18,0,0,-5
%N Expansion of psi(-x)^4 / chi(-x)^2 in powers of x where psi(), chi() 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="/A187150/b187150.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^(-7/12) * (eta(q) * eta(q^4)^2 / eta(q^2))^2 in powers of q.
%F Euler transform of period 4 sequence [ -2, 0, -2, -4, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 288 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A187149.
%e G.f. = 1 - 2*x + x^2 - 2*x^3 + 4*x^5 + x^6 + 2*x^7 - 5*x^8 - 5*x^10 + ...
%e G.f. = q^7 - 2*q^19 + q^31 - 2*q^43 + 4*q^67 + q^79 + 2*q^91 - 5*q^103 + ...
%t a[ n_] := SeriesCoefficient[ (QPochhammer[ x] QPochhammer[ x^4]^2 / QPochhammer[ x^2])^2, {x, 0, n}]; (* _Michael Somos_, Sep 02 2015 *)
%t a[ n_] := SeriesCoefficient[ (1/4) x^(-1/2) EllipticTheta[ 2, Pi/4, x^(1/2)]^4 / QPochhammer[ x, x^2]^2, {x, 0, n}]; (* _Michael Somos_, Sep 02 2015 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A))^2, n))};
%Y Cf. A187149.
%K sign
%O 0,2
%A _Michael Somos_, Mar 05 2011
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