A128639 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.

1, 8, 40, 152, 488, 1392, 3640, 8896, 20584, 45512, 96816, 199200, 398072, 775216, 1475264, 2749776, 5029736, 9043344, 16005352, 27918304, 48047280, 81661504, 137183136, 227952960, 374924152, 610743224, 985891568, 1577869784, 2504850112, 3945854640, 6170415888

Offset: 0

Author

Michael Somos, Mar 16 2007

Keywords

• nonn

Comments

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

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

Examples

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A128637, A128638.

Programs

Mathematica

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 *)

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))};

Formula

Expansion of (phi(-q^3) / phi(-q))^4 in powers of q where phi() is a Ramanujan theta function.

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

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

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.

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

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

a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (2^(3/4) * 3^(9/4) * n^(3/4)). - Vaclav Kotesovec, Sep 08 2015

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