A319554 Expansion of 1/theta_4(q)^12 in powers of q = exp(Pi i t).

1, 24, 312, 2912, 21816, 139152, 783328, 3986112, 18650424, 81251896, 332798544, 1291339296, 4776117216, 16922753616, 57683178432, 189821722688, 604884735288, 1871370360240, 5633654421720, 16535803556064, 47405095227984, 132942579098368, 365211946954656

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 A286346.

a(n) = (-1)^n * A004413(n).

a(0) = 1, a(n) = (24/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)^12.

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = 8 * Gamma(3/4)^12 / Pi^3 = A389041. - 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)^12))

Crossrefs

1/theta_4(q)^b: A015128 (b=1), A001934 (b=2), A319552 (b=3), A284286 (b=4), A319553 (b=8), this sequence (b=12).

Cf. A002131, A002448 (theta_4(q)), A004413, A286346.

Sequence in context: A053215 A290939 A004413 * A396035 A069779 A393589

Adjacent sequences: A319551 A319552 A319553 * A319555 A319556 A319557

Keyword

nonn

Author

Seiichi Manyama, Sep 22 2018

Status

approved