A128639 - OEIS

%I #28 Mar 12 2021 23:54:29

%S 1,8,40,152,488,1392,3640,8896,20584,45512,96816,199200,398072,775216,

%T 1475264,2749776,5029736,9043344,16005352,27918304,48047280,81661504,

%U 137183136,227952960,374924152,610743224,985891568,1577869784,2504850112,3945854640,6170415888

%N Expansion of (1/3) * (c(q)^2 / c(q^2)) / (b(q)^2 / b(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="/A128639/b128639.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 (phi(-q^3) / phi(-q))^4 in powers of q where phi() is a Ramanujan theta function.

%F Expansion of ((eta(q^3) / eta(q))^2 * (eta(q^2) / eta(q^6)))^4 in powers of q.

%F Euler transform of period 6 sequence [ 8, 4, 0, 4, 8, 0, ...].

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u * (1-v) * (1-9*v) - (u-v)^2.

%F G.f.: (Product_{k>0} (1 + x^k + x^(2*k)) / (1 - x^k + x^(2*k)) )^4.

%F a(n) = 8 * A128638(n) unless n = 0. Convolution inverse of A128637.

%F a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (2^(3/4) * 3^(9/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 08 2015

%F Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = 1/3 + (1/9)*sqrt(3) + (1/9)*sqrt(9+6*sqrt(3)). - _Simon Plouffe_, Mar 02 2021

%e G.f. = 1 + 8*q + 40*q^2 + 152*q^3 + 488*q^4 + 1392*q^5 + 3640*q^6 + ...

%t nmax = 40; CoefficientList[Series[Product[((1 + x^k + x^(2*k)) / (1 - x^k + x^(2*k)))^4, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 08 2015 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( ((eta(x^3 + A) / eta(x + A))^2 * eta(x^2 + A) / eta(x^6 + A))^4, n))};

%Y Cf. A128637, A128638.

%K nonn

%O 0,2

%A _Michael Somos_, Mar 16 2007