A208384 Expansion of c(q) * b(q^2) * c(q^2) / 9 in powers of q where b(), c() are cubic AGM theta functions.

1, 1, 0, -2, -3, 0, -4, -2, 0, 6, 12, 0, 8, -4, 0, 4, -9, 0, -16, 6, 0, -24, -12, 0, 7, 8, 0, 8, 3, 0, 44, 4, 0, 18, 12, 0, -34, -16, 0, -12, 33, 0, -40, -24, 0, 24, -60, 0, -33, 7, 0, -16, 27, 0, 72, 8, 0, -6, 24, 0, 50, 44, 0, -8, -24, 0, 8, 18, 0, -24

Offset

1,4

Comments

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Links

Seiichi Manyama, Table of n, a(n) for n = 1..1000

N. Elkies, Complexity of computing expansion of a newform level 18 weight 3 and character [3] - OEIS A116418

Formula

Expansion of eta(q^2)^2 * eta(q^3)^3 * eta(q^6)^2 / eta(q) in powers of q.

Euler transform of period 6 sequence [ 1, -1, -2, -1, 1, -6, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = 8748^(1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A116418.

G.f.: x * Product_{k>0} (1 - x^(2*k))^2 * (1 - x^(3*k))^3 * (1 - x^(6*k))^2 / (1 - x^k).

a(3*n) = 0. a(3*n + 1) = A116418(n). a(3*n + 2) = A122407(n).

Example

G.f. = q + q^2 - 2*q^4 - 3*q^5 - 4*q^7 - 2*q^8 + 6*q^10 + 12*q^11 + 8*q^13 + ...

Mathematica

a[ n_] := SeriesCoefficient[ q QPochhammer[ q^2]^2 QPochhammer[ q^3]^3 QPochhammer[ q^6]^2 / QPochhammer[ q], {q, 0, n}]; (* Michael Somos, Mar 30 2015 *)

Prog

(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A)^3 * eta(x^6 + A)^2 / eta(x + A), n))};

Crossrefs

Cf. A116418, A122407.

Sequence in context: A025639 A195827 A246997 * A327800 A286236 A230451

Adjacent sequences: A208381 A208382 A208383 * A208385 A208386 A208387

Keyword

sign

Author

Michael Somos, Feb 25 2012

Status

approved