A284286 Expansion of eta(q^2)^4 / eta(q)^8 in powers of q.

1, 8, 40, 160, 552, 1712, 4896, 13120, 33320, 80872, 188784, 425952, 932640, 1988080, 4137024, 8422848, 16810536, 32943760, 63482760, 120440608, 225217904, 415498496, 756920160, 1362645440, 2425895712, 4273590392, 7454092720, 12879684160, 22056267840

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Formula

a(n) = (-1)^n * A004405(n).

a(0) = 1, a(n) = (8/n)*Sum_{k=1..n} A002131(k)*a(n-k) for n > 0.

G.f.: Prod_{k>0} (1 - x^(2k))^4 / (1 - x^k)^8.

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = 2 * Gamma(3/4)^4 / Pi = A389019. - Simon Plouffe, Sep 22 2025

Mathematica

eta = QPochhammer;
CoefficientList[eta[q^2]^4/eta[q]^8 + O[q]^30, q] (* Jean-François Alcover, Feb 21 2021 *)

Prog

(Julia) # JacobiTheta4 is defined in A002448.
A284286List(len) = JacobiTheta4(len, -4)
A284286List(29) |> println # Peter Luschny, Mar 12 2018

Crossrefs

Column k=4 of A288515.

Cf. A002131, A004405, A096727.

Sequence in context: A341365 A394656 A004405 * A001789 A074412 A364619

Adjacent sequences: A284283 A284284 A284285 * A284287 A284288 A284289

Keyword

nonn

Author

Seiichi Manyama, May 02 2017

Status

approved