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%I #23 Feb 12 2018 02:52:17
%S 1,-9,54,-324,1989,-12204,74844,-459072,2815830,-17271468,105938118,
%T -649793448,3985642908,-24446767374,149949318096,-919745243064,
%U 5641448209173,-34602992662356,212244632371188,-1301846473509156,7985145356345268,-48978545212087776
%N Expansion of b(q) / a(q) in powers of q of cubic AGM theta function.
%H G. C. Greubel, <a href="/A115784/b115784.txt">Table of n, a(n) for n = 0..1000</a>
%H J. M. Borwein, P. B. Borwein and F. Garvan, <a href="http://dx.doi.org/10.1090/S0002-9947-1994-1243610-6">Some Cubic Modular Identities of Ramanujan</a>, Trans. Amer. Math. Soc. 343 (1994), 35-47.
%F Expansion of eta(q)^3 / (eta(q)^3 + 9 * eta(q^9)^3) in powers of q.
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (1 - u*v)^3 - (1 - u^3) * (1 - v^3).
%F G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (1 + 2*u)^3 * v^3 - 9 * u * (1 + u + u^2).
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = (1 + 2*u1) * (1 + 2*u2) * u3*u6 - 3 * (u1 + u2 + u1*u2).
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = (1/3) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A058091.
%F G.f.: 1 / (1 + 9 * x * (Product_{k>0} (1 - x^(9*k)) / (1 - x^k))^3).
%F Convolution inverse is A215690. Convolution with A004016 is A005928.
%F a(n) ~ (-1)^n * 8 * sqrt(3) * Pi^(5/2) * exp(Pi*n/sqrt(3)) / Gamma(1/6)^3. - _Vaclav Kotesovec_, Nov 14 2015
%e 1 - 9*q + 54*q^2 - 324*q^3 + 1989*q^4 - 12204*q^5 + 74844*q^6 - 459072*q^7 + ...
%t QP = QPochhammer; s = QP[q]^3/(QP[q]^3 + 9*q*QP[q^9]^3) + O[q]^30; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 14 2015, adapted from PARI *)
%t eta[q_] := q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[eta[q]^3/ (eta[q]^3 + 9*eta[q^9]^3), {q,0,n}]; Table[a[n], {n,0,50}] (* _G. C. Greubel_, Feb 11 2018 *)
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 / (eta(x + A)^3 + 9 * x * eta(x^9 + A)^3), n))}
%Y Cf. A004016, A005928, A215690, A258942.
%K sign
%O 0,2
%A _Michael Somos_, Jan 31 2006
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