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A121596 Expansion of q^(-1/2)(eta(q^3)/eta(q))^6 in powers of q.

%I A121596 #9 Nov 02 2018 12:35:42
%S A121596 1,6,27,92,279,756,1913,4536,10260,22220,46479,94176,185749,357426,
%T A121596 673056,1242404,2252772,4017816,7058609,12228060,20911230,35330324,
%U A121596 59023728,97568712,159693831,258941124,416181510,663337512,1048935414
%N A121596 Expansion of q^(-1/2)(eta(q^3)/eta(q))^6 in powers of q.
%H A121596 G. C. Greubel, <a href="/A121596/b121596.txt">Table of n, a(n) for n = 0..1000</a>
%F A121596 Euler transform of period 3 sequence [ 6, 6, 0, ...].
%F A121596 Given g.f. A(x), then B(x)=x*A(x)^2 satisfies 0=f(B(x), B(x^2)) where f(u,v)=u^3+v^3-u*v-24*u*v*(u+v)-729*u^2*v^2.
%F A121596 G.f.: (Product_{k>0} (1-x^(3*k))/(1-x^k))^6.
%F A121596 a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (27 * 2^(3/4) * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 07 2015
%t A121596 nmax = 40; CoefficientList[Series[Product[((1-x^(3*k)) / (1-x^k))^6, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 07 2015 *)
%t A121596 CoefficientList[Series[(QPochhammer[q^3]/QPochhammer[q])^6, {q,0,50}], q] (* _G. C. Greubel_, Nov 02 2018 *)
%o A121596 (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( (eta(x^3+A)/eta(x+A))^6, n))}
%K A121596 nonn
%O A121596 0,2
%A A121596 _Michael Somos_, Aug 09 2006