%I #19 Mar 12 2021 22:24:46
%S 1,-2,1,-2,-4,12,-3,10,-3,-20,-7,-8,29,-10,25,-28,-12,54,20,34,-74,
%T -42,-80,22,53,40,-43,16,73,-50,114,-38,-20,-68,104,-100,-47,114,-47,
%U -24,-100,-68,-151,50,137,244,-40,326,-23,-194,-30,50,-100,-160,6,-274
%N Expansion of psi(-x)^2 * f(-x^4)^6 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 G. C. Greubel, <a href="/A225564/b225564.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^(-5/4) (eta(q) * eta(q^4)^4 / eta(q^2))^2 in powers of q.
%F Expansion of chi(-x)^2 * f(-x^4)^8 = psi(-x)^8 / chi(-x)^6 in powers of x where psi(), chi(), f() are Ramanujan theta functions.
%F Euler transform of period 4 sequence [ -2, 0, -2, -8, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 1024 (t / i)^6 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A215600.
%F G.f.: Product_{k>0} (1 - x^(4*k))^8 * (1 - x^(2*k - 1))^2.
%e 1 - 2*x + x^2 - 2*x^3 - 4*x^4 + 12*x^5 - 3*x^6 + 10*x^7 - 3*x^8 - 20*x^9 + ...
%e q^5 - 2*q^9 + q^13 - 2*q^17 - 4*q^21 + 12*q^25 - 3*q^29 + 10*q^33 - 3*q^37 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ q^4]^8 QPochhammer[ q, q^2]^2, {q, 0, n}]
%t a[ n_] := SeriesCoefficient[ (1/ 16) EllipticTheta[ 2, Pi/4, q^(1/2)]^8 / QPochhammer[ q, q^2]^6, {q, 0, n + 1}]
%t a[ n_] := SeriesCoefficient[ (1/2) QPochhammer[ q^4]^6 EllipticTheta[ 2, Pi/4, q^(1/2)]^2, {q, 0, n + 1/4}]
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A)^4 / eta(x^2 + A))^2, n))}
%Y Cf. A215600.
%K sign
%O 0,2
%A _Michael Somos_, May 17 2013