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A217771 Expansion of (phi(-x) / phi(-x^3))^2 in powers of x where phi() is a Ramanujan theta function.

%I A217771 #24 Feb 16 2025 08:33:18
%S A217771 1,-4,4,4,-12,8,12,-32,20,28,-72,48,60,-152,96,120,-300,184,228,-560,
%T A217771 344,416,-1008,608,732,-1756,1048,1252,-2976,1768,2088,-4928,2900,
%U A217771 3408,-7992,4672,5460,-12728,7408,8600,-19944,11544,13344,-30800,17744,20424
%N A217771 Expansion of (phi(-x) / phi(-x^3))^2 in powers of x where phi() is a Ramanujan theta function.
%C A217771 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A217771 G. C. Greubel, <a href="/A217771/b217771.txt">Table of n, a(n) for n = 0..1000</a>
%H A217771 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A217771 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A217771 Expansion of eta(q)^4 * eta(q^6)^2 / (eta(q^2)^2 * eta(q^3)^4) in powers of q.
%F A217771 Euler transform of period 6 sequence [ -4, -2, 0, -2, -4, 0, ...].
%F A217771 G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (1 + u) * (u + v^2) - 4 * u.
%F A217771 G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u * (3 + u * v)^2 - v * (3*u + v)^2.
%F A217771 G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 3 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A217786.
%F A217771 a(n) = - 4 * A123649(n) unless n=0.
%F A217771 Convolution inverse of A186924. Convolution square of A139137.
%e A217771 G.f. = 1 - 4*x + 4*x^2 + 4*x^3 - 12*x^4 + 8*x^5 + 12*x^6 - 32*x^7 + 20*x^8 + ...
%t A217771 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q]^2 / EllipticTheta[ 4, 0, q^3]^2, {q, 0, n}]; (* _Michael Somos_, Mar 24 2013 *)
%o A217771 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 * eta(x^6 + A)^2 / (eta(x^2 + A)^2 * eta(x^3 + A)^4), n))}
%Y A217771 Cf. A123649, A186924, A217786.
%K A217771 sign
%O A217771 0,2
%A A217771 _Michael Somos_, Mar 24 2013