A128642 Expansion of (b(q) / b(q^2))^3 in powers of q where b() is a cubic AGM theta function.

1, -9, 36, -90, 180, -351, 684, -1260, 2196, -3735, 6264, -10260, 16380, -25749, 40032, -61344, 92628, -138348, 204804, -300204, 435672, -627147, 896400, -1271808, 1791324, -2507013, 3488472, -4826070, 6638688, -9085176, 12373992, -16773876, 22633812

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

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

Euler transform of period 6 sequence [ -9, 0, -6, 0, -9, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u * (1-v) * (8+u) - (u-v)^2.

G.f.: (Product_{k>0} (1 + x^(3*k)) / (1 + x^k)^3)^3.

G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 8 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A123633.

9*A128640(n) = -a(n) unless n = 0. Convolution inverse of A128643.

Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = -2 - 2*3^(1/2) + 2*6^(1/2)*3^(1/4). - Simon Plouffe, Mar 04 2021

Example

G.f. = 1 - 9*q + 36*q^2 - 90*q^3 + 180*q^4 - 351*q^5 + 684*q^6 - 1260*q^7 + ...

Mathematica

eta[q_]:= q^(1/24)*QP[q]; a[n_]:= SeriesCoefficient[((eta[q]/eta[q^2])^3*(eta[q^6]/eta[q^3]))^3, {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Mar 29 2019 *)

Prog

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

Crossrefs

Cf. A123633, A128640, A128643.

Sequence in context: A045851 A173243 A162258 * A342252 A226253 A340965

Adjacent sequences: A128639 A128640 A128641 * A128643 A128644 A128645

Keyword

sign,changed

Author

Michael Somos, Mar 16 2007

Status

approved