A233698 Expansion of b(q^2) * c(q^2) / (3 * b(q)^2) in powers of q where b(), c() are cubic AGM functions.

1, 6, 25, 84, 248, 666, 1662, 3912, 8774, 18894, 39289, 79248, 155612, 298338, 559812, 1030224, 1862647, 3313494, 5807096, 10037796, 17129888, 28886052, 48170178, 79492824, 129900206, 210314976, 337545438, 537278124, 848509124, 1330069554, 2070183912

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

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.

Formula

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

Euler transform of period 6 sequence [ 6, 4, 4, 4, 6, 0, ...].

a(n) = (-1)^n * A164271(n). 2 * a(n) = A132977(2*n + 1). -3 * a(n) = A233670(6*n + 4).

a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (2^(11/4) * 3^(9/4) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015

Example

G.f. = 1 + 6*x + 25*x^2 + 84*x^3 + 248*x^4 + 666*x^5 + 1662*x^6 + 3912*x^7 + ...
G.f. = q^2 + 6*q^5 + 25*q^8 + 84*q^11 + 248*q^14 + 666*q^17 + 1662*q^20 + ...

Mathematica

nmax=60; CoefficientList[Series[Product[((1-x^(2*k)) * (1-x^(3*k)) * (1-x^(6*k)) / (1-x^k)^3)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 13 2015 *)
eta[q_]:= q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q^(-2/3) *(eta[q^2]*eta[q^3]*eta[q^6]/eta[q]^3)^2, {q, 0, 50}], q] (* G. C. Greubel, Aug 07 2018 *)

Prog

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

Crossrefs

Cf. A132977, A164271, A233670.

Sequence in context: A256859 A133714 A164271 * A230723 A220275 A055585

Adjacent sequences: A233695 A233696 A233697 * A233699 A233700 A233701

Keyword

nonn

Author

Michael Somos, Dec 14 2013

Status

approved