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

0,2

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).

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

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

Example

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

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 *)

Prog

(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))};

Crossrefs

Cf. A128637, A128638.

Sequence in context: A028596 A264602 A125198 * A341365 A004405 A284286

Adjacent sequences: A128636 A128637 A128638 * A128640 A128641 A128642

Keyword

nonn

Author

Michael Somos, Mar 16 2007

Status

approved