A212484 - OEIS

%I #23 Aug 12 2018 00:59:11

%S 1,1,3,6,11,18,30,48,75,114,170,252,366,526,744,1044,1451,1998,2730,

%T 3700,4986,6672,8876,11736,15438,20207,26322,34134,44072,56682,72612,

%U 92680,117867,149400,188758,237744,298554,373838,466836,581412,722266,895014

%N 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.

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

%H G. C. Greubel, <a href="/A212484/b212484.txt">Table of n, a(n) for n = 0..2500</a> (terms 0..42 from Michael Somos)

%H Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015

%F 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.

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

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

%F a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (2^(11/4) * sqrt(3) * n^(3/4)). - _Vaclav Kotesovec_, Oct 13 2015

%e 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 + ...

%t 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 *)

%t 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 *)

%o (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))};

%Y Cf. A123629.

%K nonn

%O 0,3

%A _Michael Somos_, Jun 02 2012