A128636 - OEIS

%I #25 Mar 12 2021 23:54:21

%S 1,1,-3,3,5,-18,15,24,-75,57,86,-252,183,262,-744,522,725,-1998,1365,

%T 1852,-4986,3336,4436,-11736,7719,10103,-26322,17067,22040,-56682,

%U 36306,46336,-117867,74700,94378,-237744,149277,186926,-466836,290706,361126,-895014

%N 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 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

%H G. C. Greubel, <a href="/A128636/b128636.txt">Table of n, a(n) for n = 0..1000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F 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 Expansion of (eta(q^6) / eta(q)) * (eta(q^2) / eta(q^3))^5 in powers of q.

%F Euler transform of period 6 sequence [ 1, -4, 6, -4, 1, 0, ...].

%F 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 G.f.: Product_{k>0} (1 - x^(6*k)) / (1 - x^k) * ((1 - x^(2*k)) / (1 - x^(3*k)))^5.

%F A123633(n) = a(n) unless n = 0. Convolution inverse of A128641.

%F 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 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 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 (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 Cf. A123633, A128641.

%K sign

%O 0,3

%A _Michael Somos_, Mar 16 2007