|
%I #16 Mar 12 2021 22:24:44
%S 1,3,6,12,24,45,78,132,222,363,576,900,1392,2121,3180,4716,6936,10098,
%T 14550,20796,29520,41595,58176,80856,111750,153561,209820,285240,
%U 385968,519840,696960,930516,1237470,1639314,2163456,2845080,3728904,4871211
%N Expansion of psi(-q^3) / psi(-q)^3 in powers of q where psi() is a Ramanujan theta function.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A132974/b132974.txt">Table of n, a(n) for n = 0..1000</a>
%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
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of eta(q^2)^3 * eta(q^3) * eta(q^12) / (eta(q)^3 * eta(q^4)^3 * eta(q^6) ) in powers of q.
%F Euler transform of period 12 sequence [3, 0, 2, 3, 3, 0, 3, 3, 2, 0, 3, 2, ...].
%F G.f.: Product_{k>0} (1 - x^(3*k)) * (1 + x^(6*k)) / ( (1 - x^k) * (1 + x^(2*k)) )^3.
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (108)^(-1/2) (t/i)^(-1) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A133637.
%F A132979(n) = (-1)^n * a(n). Convolution inverse of A132973.
%F a(n) ~ exp(2*Pi*sqrt(n/3)) / (2 * 3^(5/4) * n^(5/4)). - _Vaclav Kotesovec_, Oct 13 2015
%e G.f. = 1 + 3*q + 6*q^2 + 12*q^3 + 24*q^4 + 45*q^5 + 78*q^6 + 132*q^7 + ...
%t a[ n_] := SeriesCoefficient[ 2 EllipticTheta[ 2, Pi/4, q^(3/2)] / EllipticTheta[ 2, Pi/4, q^(1/2)]^3 , {q, 0, n}]; (* _Michael Somos_, Sep 26 2017 *)
%t nmax=60; CoefficientList[Series[Product[(1-x^(3*k)) * (1+x^(6*k)) / ( (1-x^k)^3 * (1+x^(2*k))^3 ),{k,1,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Oct 13 2015 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^3 + A) * eta(x^12 + A ) / (eta(x + A)^3 * eta(x^4 + A)^3 * eta(x^6 + A)), n))};
%Y Cf. A132973, A132979.
%K nonn
%O 0,2
%A _Michael Somos_, Sep 07 2007
|
