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%I #11 Mar 12 2021 22:24:44
%S 1,-1,0,2,-3,0,4,-6,0,10,-12,0,20,-24,0,36,-45,0,64,-78,0,112,-132,0,
%T 189,-222,0,308,-363,0,492,-576,0,778,-900,0,1210,-1392,0,1844,-2121,
%U 0,2776,-3180,0,4144,-4716,0,6114,-6936,0,8914,-10098,0,12884,-14550
%N Expansion of q^(-1) * psi(-q) / psi(-q^3)^3 in powers of q where psi() is a Ramanujan theta function.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H G. C. Greubel, <a href="/A133637/b133637.txt">Table of n, a(n) for n = -1..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 (3 * c(q^2)) / (c(q) * c(q^4)) in powers of q where c() is a cubic AGM function.
%F Expansion of eta(q) * eta(q^4) * eta(q^6)^3 / (eta(q^2) * eta(q^3)^3 * eta(q^12)^3) in powers of q.
%F Euler transform of period 12 sequence [ -1, 0, 2, -1, -1, 0, -1, -1, 2, 0, -1, 2, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (3/4)^(1/2) (t/i)^(-1) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A132974.
%F a(3*n + 1) = 0. a(3*n) = - A132974(n).
%F Convolution inverse of A113427.
%F a(3*n - 1) = A263993(n). - _Michael Somos_, Oct 31 2015
%e G.f. = 1/q - 1 + 2*q^2 - 3*q^3 + 4*q^5 - 6*q^6 + 10*q^8 - 12*q^9 + 20*q^11 - ...
%t a[ n_] := SeriesCoefficient[ 2 EllipticTheta[ 2, Pi/4, q^(1/2)] / EllipticTheta[ 2, Pi/4, q^(3/2)]^3, {q, 0, n}]; (* _Michael Somos_, Oct 31 2015 *)
%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^3 / (eta(x^2 + A) * eta(x^3 + A)^3 * eta(x^12 + A)^3), n))};
%Y Cf. A113427, A132974, A263993.
%K sign
%O -1,4
%A _Michael Somos_, Sep 18 2007
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