A212484 Expansion of c(q^2) * b(q^6) / (b(q) * c(q) * b(q^3) * c(q^3))^(1/2) in powers of q where b(), c() are cubic AGM theta functions.

1, 1, 3, 6, 11, 18, 30, 48, 75, 114, 170, 252, 366, 526, 744, 1044, 1451, 1998, 2730, 3700, 4986, 6672, 8876, 11736, 15438, 20207, 26322, 34134, 44072, 56682, 72612, 92680, 117867, 149400, 188758, 237744, 298554, 373838, 466836, 581412, 722266, 895014

Offset: 0

Author

Michael Somos, Jun 02 2012

Keywords

• nonn

Comments

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

Examples

			G.f. = 1 + q + 3*q^2 + 6*q^3 + 11*q^4 + 18*q^5 + 30*q^6 + 48*q^7 + 75*q^8 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2500 (terms 0..42 from Michael Somos)
• Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015

Crossrefs

Cf. A123629.

Programs

Mathematica

nmax=60; CoefficientList[Series[Product[(1-x^(6*k))^6 / ((1-x^k) * (1-x^(2*k)) * (1-x^(3*k))^2 * (1-x^(9*k)) * (1-x^(18*k))),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Oct 13 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -q^3]^2 QPochhammer[ q^12]^2 / (QPochhammer[ q] QPochhammer[ q^2] QPochhammer[ q^9] QPochhammer[ q^18]), {q, 0, n}]; (* Michael Somos, Oct 24 2015 *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A)^6 / (eta(x + A) * eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^9 + A) * eta(x^18 + A)), n))};

Formula

Expansion of eta(q^6)^6 / (eta(q) * eta(q^2) * eta(q^3)^2 * eta(q^9) * eta(q^18)) in powers of q.

Euler transform of period 18 sequence [1, 2, 3, 2, 1, -2, 1, 2, 4, 2, 1, -2, 1, 2, 3, 2, 1, 0, ...].

a(n) = A123629(n) unless n=0.

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