A128641 Expansion of (1/3) * (c(q)^2 / c(q^2)) / (b(q^2)^2 / b(q)) in powers of q where b(), c() are cubic AGM theta functions.

1, -1, 4, -10, 20, -39, 76, -140, 244, -415, 696, -1140, 1820, -2861, 4448, -6816, 10292, -15372, 22756, -33356, 48408, -69683, 99600, -141312, 199036, -278557, 387608, -536230, 737632, -1009464, 1374888, -1863764, 2514868, -3378948, 4521672, -6027000, 8002676

Offset

0,3

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)^3 / phi(-q)) / (psi(q)^3 / psi(q^3)) in powers of q where phi(), psi() are Ramanujan theta functions.

Expansion of (eta(q) / eta(q^6)) * (eta(q^3) / eta(q^2))^5 in powers of q.

Euler transform of period 6 sequence [ -1, 4, -6, 4, -1, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u * (1-v) * (8-9*u) + (u-v)^2.

G.f.: Product_{k>0} (1 - x^k) / (1 - x^(6*k)) * ((1 - x^(3*k)) / (1 - x^(2*k)))^5.

A128640(n) = -a(n) unless n = 0. Convolution inverse of A128636.

a(n) ~ (-1)^n * exp(2*Pi*sqrt(n/3)) / (2 * 3^(9/4) * n^(3/4)). - Vaclav Kotesovec, Jun 06 2018

Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = 2/3 - (2/9)*sqrt(3) + (2/9)*sqrt(6)*3^(1/4). - Simon Plouffe, Mar 04 2021

a(n) = (-1)^n*A164617(n). - Michael Somos, Apr 24 2023

Example

G.f. = 1 - q + 4*q^2 - 10*q^3 + 20*q^4 - 39*q^5 + 76*q^6 - 140*q^7 + ...

Mathematica

eta[x_] := x^(1/24)*QPochhammer[x]; A128641[n_] := SeriesCoefficient[ (eta[q]/eta[q^6])*(eta[q^3]/eta[q^2])^5, {q, 0, n}]; Table[A128641[n], {n, 0, 50}] (* G. C. Greubel, Aug 22 2017 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^3 + A) / eta(x^2 + A))^5 * eta(x + A) / eta(x^6 + A), n))};

Crossrefs

Cf. A128636, A128640, A164617.

Sequence in context: A261635 A258346 A128640 * A164617 A366646 A038421

Adjacent sequences: A128638 A128639 A128640 * A128642 A128643 A128644

Keyword

sign

Author

Michael Somos, Mar 16 2007

Status

approved