%I #16 Feb 16 2025 08:33:14
%S 1,-4,4,0,2,0,-8,0,-5,16,4,0,-10,0,-8,0,9,-8,0,0,14,0,16,0,-10,-32,4,
%T 0,0,0,8,0,14,20,-20,0,2,0,0,0,-11,16,-20,0,-32,0,16,0,0,40,4,0,14,0,
%U -8,0,-9,-32,-20,0,26,0,0,0,2,-36,28,0,0,0,16,0,16,0,28,0,-22,0,0,0,14,-56,-16
%N Expansion of psi(-x)^4 * chi(-x^2)^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="/A187149/b187149.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="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of q^(-1/3) * (eta(q)^2 * eta(x^4) / eta(x^2))^2 in powers of q.
%F Euler transform of period 4 sequence [ -4, -2, -4, -4, ...].
%F a(n) = (-1)^n * A106508(n). a(4*n + 3) = a(8*n + 5) = 0.
%F G.f.: Product_{k>0} (1 + (-x)^k)^4 * (1 - x^(2*k))^4 / (1 + x^(2*k))^2.
%e G.f. = 1 - 4*x + 4*x^2 + 2*x^4 - 8*x^6 - 5*x^8 + 16*x^9 + 4*x^10 - 10*x^12 + ...
%e G.f. = q - 4*q^4 + 4*q^7 + 2*q^13 - 8*q^19 - 5*q^25 + 16*q^28 + 4*q^31 + ...
%t a[ n_] := SeriesCoefficient[ (QPochhammer[ x]^2 QPochhammer[ x^4] / 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^2, x^4]^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)^2 * eta(x^4 + A) / eta(x^2 + A))^2, n))};
%Y Cf. A106508.
%K sign,changed
%O 0,2
%A _Michael Somos_, Mar 05 2011