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A128636 Expansion of 3 * (b(q^2)^2 / b(q)) / (c(q)^2 / c(q^2)) in powers of q where b(), c() are cubic AGM theta functions.

%I A128636 #26 Feb 16 2025 08:33:05
%S A128636 1,1,-3,3,5,-18,15,24,-75,57,86,-252,183,262,-744,522,725,-1998,1365,
%T A128636 1852,-4986,3336,4436,-11736,7719,10103,-26322,17067,22040,-56682,
%U A128636 36306,46336,-117867,74700,94378,-237744,149277,186926,-466836,290706,361126,-895014
%N A128636 Expansion of 3 * (b(q^2)^2 / b(q)) / (c(q)^2 / c(q^2)) in powers of q where b(), c() are cubic AGM theta functions.
%C A128636 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A128636 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H A128636 G. C. Greubel, <a href="/A128636/b128636.txt">Table of n, a(n) for n = 0..1000</a>
%H A128636 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A128636 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A128636 Expansion of (psi(q)^3 / psi(q^3)) / (phi(-q^3)^3 / phi(-q)) in powers of q where phi(), psi() are Ramanujan theta functions.
%F A128636 Expansion of (eta(q^6) / eta(q)) * (eta(q^2) / eta(q^3))^5 in powers of q.
%F A128636 Euler transform of period 6 sequence [ 1, -4, 6, -4, 1, 0, ...].
%F A128636 G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v* (1-v)* (9-8*u) + (u-v)^2.
%F A128636 G.f.: Product_{k>0} (1 - x^(6*k)) / (1 - x^k) * ((1 - x^(2*k)) / (1 - x^(3*k)))^5.
%F A128636 A123633(n) = a(n) unless n = 0. Convolution inverse of A128641.
%F A128636 Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = -(3/8)*sqrt(3) + (3/8)*sqrt(9 + 6*sqrt(3)). - _Simon Plouffe_, Mar 02 2021
%e A128636 G.f. = 1 + q - 3*q^2 + 3*q^3 + 5*q^4 - 18*q^5 + 15*q^6 + 24*q^7 - 75*q^8 + ...
%t A128636 eta[x_] := x^(1/24)*QPochhammer[x]; A128636[n_] := SeriesCoefficient[(eta[q^6]/eta[q])*(eta[q^2]/eta[q^3])^5, {q, 0, n}]; Table[A128636[n], {n, 0, 50}] (* _G. C. Greubel_, Aug 21 2017 *)
%o A128636 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) / eta(x^3 + A))^5 * eta(x^6 + A) / eta(x + A), n))};
%Y A128636 Cf. A123633, A128641.
%K A128636 sign
%O A128636 0,3
%A A128636 _Michael Somos_, Mar 16 2007