A258111 Expansion of b(q^3) * b(q^12) / (b(-q) * b(q^6)) in powers of q where b() is a cubic AGM theta function.

1, -3, 9, -24, 57, -126, 264, -528, 1017, -1896, 3438, -6084, 10536, -17898, 29880, -49104, 79545, -127170, 200856, -313692, 484830, -742080, 1125540, -1692648, 2525160, -3738765, 5496246, -8025432, 11643576, -16790310, 24072048, -34321560, 48677625

Offset

0,2

Comments

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

Formula

Expansion of eta(q)^3 * eta(q^3)^2 * eta(q^4)^3 * eta(q^12)^2 * eta(q^18) / (eta(q^2)^9 * eta(q^9) * eta(q^36)) in powers of q.

Euler transform of period 36 sequence [ -3, 6, -5, 3, -3, 4, -3, 3, -4, 6, -3, -1, -3, 6, -5, 3, -3, 4, -3, 3, -5, 6, -3, -1, -3, 6, -4, 3, -3, 4, -3, 3, -5, 6, -3, 0, ...]. - Corrected by Sean A. Irvine, Mar 06 2020

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A164615.

Convolution inverse of A258108.

a(n) ~ (-1)^n * exp(4*Pi*sqrt(n)/3) / (2^(5/2) * 3^(3/2) * n^(3/4)). - Vaclav Kotesovec, Jun 06 2018

Example

G.f. = 1 - 3*q + 9*q^2 - 24*q^3 + 57*q^4 - 126*q^5 + 264*q^6 - 528*q^7 + ...

Mathematica

QP = QPochhammer; A258111[n_] := SeriesCoefficient[(QP[q]^3*QP[q^3]^2 *QP[q^4]^3*QP[q^12]^2*QP[q^18])/(QP[q^2]^9*QP[q^9]*QP[q^36]), {q, 0, n}]; Table[A258111[n], {n, 0, 50}] (* G. C. Greubel, Oct 18 2017 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^3 + A)^2 * eta(x^4 + A)^3 * eta(x^12 + A)^2 * eta(x^18 + A) / (eta(x^2 + A)^9 * eta(x^9 + A) * eta(x^36 + A)), n))};

Crossrefs

Cf. A164615, A258108.

Sequence in context: A245762 A291706 A089830 * A360197 A109175 A120539

Adjacent sequences: A258108 A258109 A258110 * A258112 A258113 A258114

Keyword

sign

Author

Michael Somos, May 20 2015

Status

approved