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A233698 Expansion of b(q^2) * c(q^2) / (3 * b(q)^2) in powers of q where b(), c() are cubic AGM functions.

%I A233698 #13 Aug 08 2018 07:54:39
%S A233698 1,6,25,84,248,666,1662,3912,8774,18894,39289,79248,155612,298338,
%T A233698 559812,1030224,1862647,3313494,5807096,10037796,17129888,28886052,
%U A233698 48170178,79492824,129900206,210314976,337545438,537278124,848509124,1330069554,2070183912
%N A233698 Expansion of b(q^2) * c(q^2) / (3 * b(q)^2) in powers of q where b(), c() are cubic AGM functions.
%C A233698 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H A233698 G. C. Greubel, <a href="/A233698/b233698.txt">Table of n, a(n) for n = 0..2500</a>
%H A233698 Vaclav Kotesovec, <a href="https://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], 2015-2016.
%F A233698 Expansion of (eta(q^2) * eta(q^3) * eta(q^6) / eta(q)^3)^2 in powers of q.
%F A233698 Euler transform of period 6 sequence [ 6, 4, 4, 4, 6, 0, ...].
%F A233698 a(n) = (-1)^n * A164271(n). 2 * a(n) = A132977(2*n + 1). -3 * a(n) =  A233670(6*n + 4).
%F A233698 a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (2^(11/4) * 3^(9/4) * n^(3/4)). - _Vaclav Kotesovec_, Oct 13 2015
%e A233698 G.f. = 1 + 6*x + 25*x^2 + 84*x^3 + 248*x^4 + 666*x^5 + 1662*x^6 + 3912*x^7 + ...
%e A233698 G.f. = q^2 + 6*q^5 + 25*q^8 + 84*q^11 + 248*q^14 + 666*q^17 + 1662*q^20 + ...
%t A233698 nmax=60; CoefficientList[Series[Product[((1-x^(2*k)) * (1-x^(3*k)) * (1-x^(6*k)) / (1-x^k)^3)^2,{k,1,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Oct 13 2015 *)
%t A233698 eta[q_]:= q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q^(-2/3) *(eta[q^2]*eta[q^3]*eta[q^6]/eta[q]^3)^2, {q, 0, 50}], q] (* _G. C. Greubel_, Aug 07 2018 *)
%o A233698 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^3 + A) * eta(x^6 + A) / eta(x + A)^3)^2, n))}
%Y A233698 Cf. A132977, A164271, A233670.
%K A233698 nonn
%O A233698 0,2
%A A233698 _Michael Somos_, Dec 14 2013