A319553 Expansion of 1/theta_4(q)^8 in powers of q = exp(Pi i t).

1, 16, 144, 960, 5264, 25056, 106944, 418176, 1520784, 5201232, 16871648, 52252992, 155341248, 445226848, 1234726272, 3323392128, 8704504976, 22234655520, 55498917840, 135595345600, 324759439584, 763505859072, 1764050361152, 4009763323008, 8975341703616, 19800832628336

Offset

0,2

Links

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

Simon Plouffe, Numbers in the base e^Pi, arXiv:2509.15609 [math.NT], 2025. See p. 17/24, marked 210.

Formula

Convolution inverse of A035016.

a(n) = (-1)^n * A004409(n).

a(0) = 1, a(n) = (16/n)*Sum_{k=1..n} A002131(k)*a(n-k) for n > 0.

G.f.: Product_{k>=1} ((1 - x^(2k))/(1 - x^k)^2)^8.

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = 4 * Gamma(3/4)^8 / Pi^2 = A389040. - Simon Plouffe, Sep 22 2025

Prog

(PARI) N=99; x='x+O('x^N); Vec(prod(k=1, N, ((1-x^(2*k))/(1-x^k)^2)^8))

Crossrefs

1/theta_4(q)^b: A015128 (b=1), A001934 (b=2), A319552 (b=3), A284286 (b=4), this sequence (b=8), A319554 (b=12).

Cf. A002131, A002448 (theta_4(q)), A004409, A035016.

Sequence in context: A128985 A341369 A004409 * A054851 A396033 A000762

Adjacent sequences: A319550 A319551 A319552 * A319554 A319555 A319556

Keyword

nonn

Author

Seiichi Manyama, Sep 22 2018

Status

approved