A319552 Expansion of 1/theta_4(q)^3 in powers of q = exp(Pi i t).

1, 6, 24, 80, 234, 624, 1552, 3648, 8184, 17654, 36816, 74544, 147056, 283440, 535008, 990912, 1803882, 3232224, 5707624, 9943536, 17106960, 29088352, 48922320, 81438528, 134261584, 219336630, 355242288, 570675904, 909674688, 1439394192, 2261635168, 3529838208

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 A213384.

a(n) = (-1)^n * A004404(n).

a(0) = 1, a(n) = (6/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)^3.

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = 2^(3/4) * Gamma(3/4)^3 / Pi^(3/4) = A389039. - 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)^3))

Crossrefs

1/theta_4(q)^b: A015128 (b=1), A001934 (b=2), this sequence (b=3), A284286 (b=4), A319553 (b=8), A319554 (b=12).

Cf. A002131, A002448 (theta_4(q)), A004404, A213384.

Sequence in context: A350413 A361474 A004404 * A201189 A001788 A068711

Adjacent sequences: A319549 A319550 A319551 * A319553 A319554 A319555

Keyword

nonn

Author

Seiichi Manyama, Sep 22 2018

Status

approved