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A128642 Expansion of (b(q) / b(q^2))^3 in powers of q where b() is a cubic AGM theta function.

%I A128642 #26 Feb 16 2025 08:33:05
%S A128642 1,-9,36,-90,180,-351,684,-1260,2196,-3735,6264,-10260,16380,-25749,
%T A128642 40032,-61344,92628,-138348,204804,-300204,435672,-627147,896400,
%U A128642 -1271808,1791324,-2507013,3488472,-4826070,6638688,-9085176,12373992,-16773876,22633812
%N A128642 Expansion of (b(q) / b(q^2))^3 in powers of q where b() is a cubic AGM theta function.
%C A128642 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A128642 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H A128642 G. C. Greubel, <a href="/A128642/b128642.txt">Table of n, a(n) for n = 0..1000</a>
%H A128642 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A128642 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A128642 Expansion of (chi(-q)^3 / chi(-q^3))^3 in powers of q where chi() is a Ramanujan theta function.
%F A128642 Expansion of ((eta(q) / eta(q^2))^3 * (eta(q^6) / eta(q^3)))^3 in powers of q.
%F A128642 Euler transform of period 6 sequence [ -9, 0, -6, 0, -9, 0, ...].
%F A128642 G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u * (1-v) * (8+u) - (u-v)^2.
%F A128642 G.f.: (Product_{k>0} (1 + x^(3*k)) / (1 + x^k)^3)^3.
%F A128642 G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 8 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A123633.
%F A128642 9*A128640(n) = -a(n) unless n = 0. Convolution inverse of A128643.
%F A128642 Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = -2 - 2*3^(1/2) + 2*6^(1/2)*3^(1/4). - _Simon Plouffe_, Mar 04 2021
%e A128642 G.f. = 1 - 9*q + 36*q^2 - 90*q^3 + 180*q^4 - 351*q^5 + 684*q^6 - 1260*q^7 + ...
%t A128642 eta[q_]:= q^(1/24)*QP[q]; a[n_]:= SeriesCoefficient[((eta[q]/eta[q^2])^3*(eta[q^6]/eta[q^3]))^3, {q, 0, n}]; Table[a[n], {n,0,50}] (* _G. C. Greubel_, Mar 29 2019 *)
%o A128642 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( ((eta(x + A) / eta(x^2 + A))^3 * eta(x^6 + A) / eta(x^3 + A))^3, n))};
%Y A128642 Cf. A123633, A128640, A128643.
%K A128642 sign
%O A128642 0,2
%A A128642 _Michael Somos_, Mar 16 2007