A128637 Expansion of 3 * (b(q)^2/b(q^2)) / (c(q)^2/c(q^2)) in powers of q where b(), c() are cubic AGM theta functions.

1, -8, 24, -24, -40, 144, -120, -192, 600, -456, -688, 2016, -1464, -2096, 5952, -4176, -5800, 15984, -10920, -14816, 39888, -26688, -35488, 93888, -61752, -80824, 210576, -136536, -176320, 453456, -290448, -370688, 942936, -597600, -755024, 1901952, -1194216

Offset

0,2

Comments

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

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

Expansion of ((eta(q) / eta(q^3))^2 * (eta(q^6) / eta(q^2)))^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) * (9-v) - (u-v)^2.

G.f.: A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u^2*v^2 + 9*u*v - 12*u*v^2 + 30*v^2 - 108*v + 81) * u - v^3.

G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 9 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128640.

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

a(n) = -8 * A123633(n) unless n=0. Convolution inverse of A128639.

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

Example

G.f. = 1 - 8*q + 24*q^2 - 24*q^3 - 40*q^4 + 144*q^5 - 120*q^6 - 192*q^7 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q] / EllipticTheta[ 4, 0, q^3])^4, {q, 0, n}]; (* Michael Somos, Apr 06 2013 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ -q^3, q^3] / QPochhammer[ -q, q])^4 /(QPochhammer[ q^3] / QPochhammer[ q])^4, {q, 0, n}]; (* Michael Somos, Apr 06 2013 *)

Prog

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

Crossrefs

Cf. A123633, A128639, A128640.

Sequence in context: A205376 A088448 A005878 * A280708 A370531 A109272

Adjacent sequences: A128634 A128635 A128636 * A128638 A128639 A128640

Keyword

sign

Author

Michael Somos, Mar 16 2007

Status

approved