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%I #26 Jun 30 2017 11:08:40
%S 1,6,27,86,243,594,1370,2916,5967,11586,21870,39852,71052,123444,
%T 210654,352480,581013,942786,1510254,2388204,3734964,5777788,8852004,
%U 13434984,20218395,30177684,44704413,65743348,96033357,139368816,201032186,288281592,411119766
%N Expansion of (eta(q^3)^2 / (eta(q) * eta(q^9)))^6 in powers of q.
%C In Berndt and Chan (1999) denoted by h(q) in Theorem 3.1. - _Michael Somos_, Oct 20 2013
%H Seiichi Manyama, <a href="/A131985/b131985.txt">Table of n, a(n) for n = -1..10000</a>
%H B. C. Berndt and H. H. Chan, <a href="http://dx.doi.org/10.4153/CMB-1999-050-1">Ramanujan and the Modular j-invariant</a>, Canad. Math. Bull., 42 (1999), 427-440.
%F Euler transform of period 9 sequence [ 6, 6, -6, 6, 6, -6, 6, 6, 0, ...].
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u+v)^3 + u*v*(27 + 9*(u+v) - u*v).
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2 + w^2 + 12*v^2 + u*w - v^2*(u+w) + 12*v*(u+w) + 27*v.
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = f(t) where q = exp(2 Pi i t).
%F G.f.: (1/x) * (Product_{k>0} (1 - x^(3*k))^2 / ((1 - x^k) * (1 - x^(9*k))))^6.
%F a(n) = A007266(n) = A045491(n) unless n=0.
%F a(n) = A131986(n) + 27 * A121589(n) unless n=0. - _Michael Somos_, Oct 20 2013
%F Convolution inverse of A121592. - _Michael Somos_, Oct 20 2013
%F a(n) ~ exp(4*Pi*sqrt(n)/3) / (sqrt(6) * n^(3/4)). - _Vaclav Kotesovec_, Sep 07 2015
%e G.f. = 1/q + 6 + 27*q + 86*q^2 + 243*q^3 + 594*q^4 + 1370*q^5 + 2916*q^6 +...
%t a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ q^3]^2 / (QPochhammer[ q] QPochhammer[ q^9]))^6, {q, 0, n}]; (* _Michael Somos_, Oct 20 2013 *)
%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^3 + A)^2 / (eta(x + A) * eta(x^9 + A)))^6, n))};
%Y Cf. A007266, A045491, A121589, A131986.
%K nonn
%O -1,2
%A _Michael Somos_, Aug 04 2007
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