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A128758 Expansion of q^(-1/3) * (eta(q^3) / eta(q))^4 in powers of q.

%I A128758 #13 Jul 04 2018 13:26:40
%S A128758 1,4,14,36,89,196,416,828,1600,2972,5390,9504,16436,27828,46364,75960,
%T A128758 122772,195728,308430,480456,740921,1131364,1712348,2569500,3825641,
%U A128758 5652872,8294612,12089016,17508609,25204428,36076540,51355368,72725909
%N A128758 Expansion of q^(-1/3) * (eta(q^3) / eta(q))^4 in powers of q.
%C A128758 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%D A128758 O. Kolberg, The coefficients of j(tau) modulo powers of 3, Acta Univ. Bergen., Series Math., Arbok for Universitetet I Bergen, Mat.-Naturv. Serie, 1962 No. 16, pp. 1-7. See v, page 1.
%H A128758 G. C. Greubel, <a href="/A128758/b128758.txt">Table of n, a(n) for n = 0..1000</a>
%F A128758 Expansion of q^(-1/3) * (1/3) * c(q) / b(q) in powers of q where b(), c() are cubic AGM theta functions.
%F A128758 Euler transform of period 3 sequence [ 4, 4, 0, ...].
%F A128758 Given g.f. A(x), then B(q) = q*A(q^3) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (u+v)^3 - u*v * (1+3*u) * (1+3*v).
%F A128758 Given g.f. A(x), then B(q)= q*A(q^3) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = u^2 + w^2 + u*w - v - 9*v^2 * (u+w).
%F A128758 G.f.: (Product_{k>0} (1 + x^k + x^(2*k)) )^4.
%F A128758 9*a(n) = A112146(3*n + 1).
%F A128758 a(n) ~ exp(4*Pi*sqrt(n)/3) / (9*sqrt(6)*n^(3/4)). - _Vaclav Kotesovec_, Sep 07 2015
%e A128758 G.f. = 1 + 4*x + 14*x^2 + 36*x^3 + 89*x^4 + 196*x^5 + 416*x^6 + 828*x^7 + ...
%e A128758 G.f. = q + 4*q^4 + 14*q^7 + 36*q^10 + 89*q^13 + 196*q^16 + 416*q^19 + ...
%t A128758 nmax = 40; CoefficientList[Series[Product[((1-x^(3*k)) / (1-x^k))^4, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 07 2015 *)
%t A128758 eta[q_]:= q^(1/24)*QPochhammer[q]; c:= q^(1/3)*(eta[q]/eta[q^3])^4; a:= CoefficientList[Series[1/c, {q,0,60}], q]; Table[a[[n]], {n,1,50}] (* _G. C. Greubel_, Jul 04 2018 *)
%o A128758 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^3 + A) / eta(x + A))^4, n))};
%Y A128758 Cf. A112146.
%K A128758 nonn
%O A128758 0,2
%A A128758 _Michael Somos_, Mar 24 2007