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A128641 Expansion of (1/3) * (c(q)^2 / c(q^2)) / (b(q^2)^2 / b(q)) in powers of q where b(), c() are cubic AGM theta functions.

%I A128641 #25 Feb 16 2025 08:33:05
%S A128641 1,-1,4,-10,20,-39,76,-140,244,-415,696,-1140,1820,-2861,4448,-6816,
%T A128641 10292,-15372,22756,-33356,48408,-69683,99600,-141312,199036,-278557,
%U A128641 387608,-536230,737632,-1009464,1374888,-1863764,2514868,-3378948,4521672,-6027000,8002676
%N A128641 Expansion of (1/3) * (c(q)^2 / c(q^2)) / (b(q^2)^2 / b(q)) in powers of q where b(), c() are cubic AGM theta functions.
%C A128641 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A128641 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H A128641 G. C. Greubel, <a href="/A128641/b128641.txt">Table of n, a(n) for n = 0..1000</a>
%H A128641 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A128641 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A128641 Expansion of (phi(-q^3)^3 / phi(-q)) / (psi(q)^3 / psi(q^3)) in powers of q where phi(), psi() are Ramanujan theta functions.
%F A128641 Expansion of (eta(q) / eta(q^6)) * (eta(q^3) / eta(q^2))^5 in powers of q.
%F A128641 Euler transform of period 6 sequence [ -1, 4, -6, 4, -1, 0, ...].
%F A128641 G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u * (1-v) * (8-9*u) + (u-v)^2.
%F A128641 G.f.: Product_{k>0} (1 - x^k) / (1 - x^(6*k)) * ((1 - x^(3*k)) / (1 - x^(2*k)))^5.
%F A128641 A128640(n) = -a(n) unless n = 0. Convolution inverse of A128636.
%F A128641 a(n) ~ (-1)^n * exp(2*Pi*sqrt(n/3)) / (2 * 3^(9/4) * n^(3/4)). - _Vaclav Kotesovec_, Jun 06 2018
%F A128641 Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = 2/3 - (2/9)*sqrt(3) + (2/9)*sqrt(6)*3^(1/4). - _Simon Plouffe_, Mar 04 2021
%F A128641 a(n) = (-1)^n*A164617(n). - _Michael Somos_, Apr 24 2023
%e A128641 G.f. = 1 - q + 4*q^2 - 10*q^3 + 20*q^4 - 39*q^5 + 76*q^6 - 140*q^7 + ...
%t A128641 eta[x_] := x^(1/24)*QPochhammer[x]; A128641[n_] := SeriesCoefficient[ (eta[q]/eta[q^6])*(eta[q^3]/eta[q^2])^5, {q, 0, n}]; Table[A128641[n], {n, 0, 50}] (* _G. C. Greubel_, Aug 22 2017 *)
%o A128641 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^3 + A) / eta(x^2 + A))^5 * eta(x + A) / eta(x^6 + A), n))};
%Y A128641 Cf. A128636, A128640, A164617.
%K A128641 sign
%O A128641 0,3
%A A128641 _Michael Somos_, Mar 16 2007