A231947 Expansion of q^(-1/3) * a(q)^2 * c(q) / 3 in powers of q where a(), c() are cubic AGM theta functions.

1, 13, 50, 72, 170, 205, 362, 360, 601, 650, 962, 864, 1370, 1224, 1850, 1584, 2451, 2210, 2880, 2520, 3722, 3277, 4490, 3600, 5330, 4706, 6242, 5040, 6912, 6120, 8500, 6624, 9410, 7813, 10610, 8424, 11882, 10250, 12672, 10440, 14521, 12506, 16130, 12240

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..2500

Formula

Expansion of q^(-1/3) * (eta(q)^3 + 9 * eta(q^9)^3)^2 * eta(q^3) / eta(q) in powers of q.

G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = 3^(1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A231948.

-9 * a(n) = A109041(3*n + 1).

Example

G.f. = 1 + 13*x + 50*x^2 + 72*x^3 + 170*x^4 + 205*x^5 + 362*x^6 + 360*x^7 + ...
G.f. = q + 13*q^4 + 50*q^7 + 72*q^10 + 170*q^13 + 205*q^16 + 362*q^19 + ...

Mathematica

eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[q^(-1/3)*(eta[q]^3 + 9*eta[q^9]^3)^2*eta[q^3]/eta[q], {q, 0, 50}], q] (* G. C. Greubel, Aug 08 2018 *)

Prog

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

Crossrefs

Cf. A109041, A231948.

Sequence in context: A274784 A294520 A189054 * A322615 A209995 A050410

Adjacent sequences: A231944 A231945 A231946 * A231948 A231949 A231950

Keyword

nonn

Author

Michael Somos, Nov 15 2013

Status

approved