cp's OEIS Frontend

A123629 Expansion of b(q^2) * c(q^6) / (b(q) * c(q^3)) in powers of q where b(), c() are cubic AGM theta functions.

%I A123629 #22 Nov 03 2022 10:15:24
%S A123629 1,3,6,11,18,30,48,75,114,170,252,366,526,744,1044,1451,1998,2730,
%T A123629 3700,4986,6672,8876,11736,15438,20207,26322,34134,44072,56682,72612,
%U A123629 92680,117867,149400,188758,237744,298554,373838,466836,581412,722266,895014
%N A123629 Expansion of b(q^2) * c(q^6) / (b(q) * c(q^3)) in powers of q where b(), c() are cubic AGM theta functions.
%C A123629 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A123629 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H A123629 G. C. Greubel, <a href="/A123629/b123629.txt">Table of n, a(n) for n = 1..1000</a>
%H A123629 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 A123629 Expansion of (eta(q^3) / eta(q^6))^2 * (eta(q^2) * eta(q^18) / (eta(q) * eta(q^9)))^3 in powers of q.
%F A123629 G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v - u*(6*v + 4*v^2).
%F A123629 Euler transform of period 18 sequence [ 3, 0, 1, 0, 3, 0, 3, 0, 4, 0, 3, 0, 3, 0, 1, 0, 3, 0, ...].
%F A123629 Convolution inverse is A123676. - _Michael Somos_, Feb 19 2015
%F A123629 Expansion of q * chi(-q^3)^2 / (chi(-q) * chi(-q^9))^3 in powers of q where chi() is a Ramanujan theta function. - _Michael Somos_, Feb 19 2015
%F A123629 G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = (1/4) * g(t) where q = exp(2 Pi i t) and g() is the g.f. for A123676. - _Michael Somos_, Feb 19 2015
%F A123629 a(3*n) = 6 * A128638(n). a(3*n + 2) = 3 * A233698(n). - _Michael Somos_, Feb 19 2015
%F A123629 a(n) ~ exp(2*sqrt(2*n)*Pi/3) / (2^(11/4)*sqrt(3)*n^(3/4)). - _Vaclav Kotesovec_, Oct 10 2015
%e A123629 G.f. = q + 3*q^2 + 6*q^3 + 11*q^4 + 18*q^5 + 30*q^6 + 48*q^7 + 75*q^8 + ...
%t A123629 nmax=60; CoefficientList[Series[Product[(1+x^k)^3 * (1+x^(9*k))^3 / (1+x^(3*k))^2,{k,1,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Oct 10 2015 *)
%t A123629 A123629[n_] := SeriesCoefficient[q*(QPochhammer[q^3]/QPochhammer[q^6])^2*(QPochhammer[q^2]*QPochhammer[q^18]/(QPochhammer[q]*QPochhammer[q^9] ))^3, {q, 0, n}]; Table[A123629[n], {n, 0, 50}] (* _G. C. Greubel_, Oct 09 2017 *)
%o A123629 (PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^3 + A) / eta(x^6 + A))^2 * (eta(x^2 + A) * eta(x^18 + A) / (eta(x + A) * eta(x^9 + A)))^3, n))};
%Y A123629 Cf. A123676, A128638, A233698.
%K A123629 nonn
%O A123629 1,2
%A A123629 _Michael Somos_, Oct 03 2006
%E A123629 Typo in xrefs corrected by _Vaclav Kotesovec_, Oct 10 2015