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%I #19 Aug 14 2018 09:01:06
%S 1,-2,-1,6,-7,-2,20,-24,-8,56,-61,-18,137,-150,-46,312,-327,-94,663,
%T -690,-199,1342,-1366,-384,2603,-2632,-739,4884,-4869,-1344,8890,
%U -8808,-2422,15784,-15487,-4212,27389,-26728,-7242,46608,-45155,-12136,77888,-75102
%N Expansion of q^(-1/3) * (eta(q) * eta(q^9))^2 / eta(q^3)^4 in powers of q.
%H G. C. Greubel, <a href="/A192329/b192329.txt">Table of n, a(n) for n = 0..2500</a>
%F Expansion of q^(-1/3) * ( (b(q) * c(q^3)) / (c(q) * b(q^3)) )^(1/2) in powers of q where b(), c() are cubic AGM functions.
%F Euler transform of period 9 sequence [ -2, -2, 2, -2, -2, 2, -2, -2, 0, ...].
%F Given g.f. A(x) then B(x) = x * A(x^3) satisfies 0 = f(B(x), B(x^2)) where f(u, v) = (u^2 - v) * (u - v^2) + 4*u^2*v^2.
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (81 t)) = f(t) where q = exp(2 Pi i t).
%F G.f.: Product_{k>0} ((1 - x^k) * (1 - x^(9*k)))^2 / (1 - x^(3*k))^4.
%F Convolution inverse of A058601.
%e G.f. = 1 - 2*x - x^2 + 6*x^3 - 7*x^4 - 2*x^5 + 20*x^6 - 24*x^7 - 8*x^8 + ...
%e G.f. = q - 2*q^4 - q^7 + 6*q^10 - 7*q^13 - 2*q^16 + 20*q^19 - 24*q^22 - 8*q^25 + ...
%t QP = QPochhammer; s = (QP[q]*QP[q^9])^2/QP[q^3]^4 + O[q]^50; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 25 2015 *)
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^9 + A))^2 / eta(x^3 + A)^4, n))}
%Y Cf. A058601.
%K sign
%O 0,2
%A _Michael Somos_, Jul 01 2011
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