|
%I #67 Sep 26 2023 17:02:01
%S 1,8,32,96,256,624,1408,3008,6144,12072,22976,42528,76800,135728,
%T 235264,400704,671744,1109904,1809568,2914272,4640256,7310592,
%U 11404416,17626944,27009024,41047992,61905088,92681664
%N Expansion of (theta_3(q) / theta_4(q))^2 in powers of q.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A. Cayley, An Elementary Treatise on Elliptic Functions, 2nd ed, 1895, p. 380, Section 488.
%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
%D N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; Eq. (34.3).
%D R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, 1922, Vol. 2, see p. 375. Eqs. (17),(18),(19).
%H T. D. Noe, <a href="/A014969/b014969.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, p. 11.
%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>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EllipticLambdaFunction.html">Elliptic Lambda Function</a>
%F Expansion of (phi(q) / phi(-q))^2 = (phi(q) / phi(-q^2))^4 = (phi(-q^2) / phi(-q))^4 = (psi(q) / psi(-q))^4 = (chi(q)^2 / chi(-q^2))^4 = (chi(q) / chi(-q))^4 = (chi(-q^2) / chi(-q)^2)^4 = (f(q) / f(-q))^4 in powers of q where phi(), psi(), chi() are Ramanujan theta functions. - _Michael Somos_, Aug 01 2011
%F Expansion of Fricke t(omega) = tau(omega) / 2 + 1 in powers of q = exp(2 Pi i omega).
%F Expansion of elliptic 1 / sqrt(1 - lambda(q)) = 1 / k'(q) in powers of the nome q = exp(Pi*i*z).
%F Euler transform of period 4 sequence [ 8, -4, 8, 0, ...]. - _Michael Somos_, Jul 07 2005
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (1 + u)^2 - 4*u*v^2. - _Michael Somos_, Nov 14 2006
%F G.f.: (theta_3(x) / theta_4(x))^2 = (Sum_{k} x^k^2) / (Sum_{k} (-x)^k^2)^2 = (Product_{k>0} (1 - x^(4*k - 2)) / ((1 - x^(4*k - 1)) * (1 - x^(4*k - 3)))^2)^4.
%F A139820(n) = (-1)^n * a(n). 8 * A107035(n) = a(n) unless n=0. 2 * A131126(n) = a(n) unless n=0. Convolution inverse of A139820.
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 1/4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A029841. - _Michael Somos_, Jun 04 2015
%F a(n) ~ exp(Pi*sqrt(2*n)) / (8 * 2^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Aug 28 2015
%F G.f.: exp(8*Sum_{k>=1} sigma(2*k - 1)*x^(2*k-1)/(2*k - 1)). - _Ilya Gutkovskiy_, Apr 19 2019
%F Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = (1/4)*sqrt(8 + 6*sqrt(2)). - _Simon Plouffe_, Mar 02 2021
%F From _Peter Bala_, Sep 25 2023: (Start)
%F G.f.: A(q) = sqrt(-lambda(-q)/lambda(q)), where lambda(q) = 16*q - 128*q^2 + 704*q^3 - 3072*q^4 + ... is the elliptic modular function in powers of the nome q = exp(i*Pi*t), the g.f. of A115977; lambda(q) = k(q)^2, where k(q) = (theta_2(q) / theta_3(q))^2 is the elliptic modulus.
%F A(q) = sqrt(G(q)), where G(q) = 1 + 16q + 128*q^2 + 704*q^3 + 3072*q^4 + ... is the g.f. of A014972. (End)
%e G.f. = 1 + 8*q + 32*q^2 + 96*q^3 + 256*q^4 + 624*q^5 + 1408*q^6 + 3008*q^7 + ...
%t a[ n_] := SeriesCoefficient[ 1 / Sqrt[1 - InverseEllipticNomeQ @ q], {q, 0, n}]; (* _Michael Somos_, Aug 01 2011 *)
%t a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] / EllipticTheta[ 4, 0, q])^2, {q, 0, n}]; (* _Michael Somos_, Aug 01 2011 *)
%t nmax=60; CoefficientList[Series[Product[((1+x^(2*k+1))/(1-x^(2*k+1)))^4,{k,0,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Aug 28 2015 *)
%t s = (QPochhammer[q^2]^3/(QPochhammer[q]^2*QPochhammer[q^4]))^4+O[q]^30; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 09 2015, adapted from PARI *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 / (eta(x + A)^2 * eta(x^4 + A)))^4, n))}; /* _Michael Somos_, Jul 07 2005 */
%Y Cf. A014972, A029841, A107035, A115977, A131126, A139820.
%K nonn,nice
%O 0,2
%A _N. J. A. Sloane_
|
