A258108 Expansion of b(-q) * b(q^6) / (b(q^3) * b(q^12)) in powers of q where b() is a cubic AGM theta function.

1, 3, 0, -3, 6, 0, -12, 15, 0, -30, 36, 0, -60, 78, 0, -117, 150, 0, -228, 276, 0, -420, 504, 0, -732, 885, 0, -1245, 1488, 0, -2088, 2454, 0, -3420, 3996, 0, -5460, 6378, 0, -8583, 9972, 0, -13344, 15378, 0, -20448, 23472, 0, -30876, 35379, 0, -46116, 52644

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^2)^9 * eta(q^9) * eta(q^36) / (eta(q)^3 * eta(q^3)^2 * eta(q^4)^3 * eta(q^12)^2 * eta(q^18)) 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, ...].

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 A164616.

a(3*n + 2) = 0. a(3*n + 1) = 3 * A132977(n). a(3*n) = -3 * A164617(n) unless n = 0.

Example

G.f. = 1 + 3*q - 3*q^3 + 6*q^4 - 12*q^6 + 15*q^7 - 30*q^9 + 36*q^10 + ...

Mathematica

QP=QPochhammer; A258108[n_] :=SeriesCoefficient[(QP[x^2]^9*QP[x^9]*QP[x^36])/(QP[x]^3*QP[x^3]^2*QP[x^4]^3*QP[x^12]^2*QP[x^18]), {x, 0, n}]; Table[A258108[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^2 + A)^9 * eta(x^9 + A) * eta(x^36 + A) / (eta(x + A)^3 * eta(x^3 + A)^2 * eta(x^4 + A)^3 * eta(x^12 + A)^2 * eta(x^18 + A)), n))};

Crossrefs

Cf. A132977, A164616, A164617.

Sequence in context: A194485 A120987 A281293 * A282610 A011076 A200278

Adjacent sequences: A258105 A258106 A258107 * A258109 A258110 A258111

Keyword

sign

Author

Michael Somos, May 20 2015

Status

approved