OFFSET
0,2
COMMENTS
REFERENCES
A. Cayley, An Elementary Treatise on Elliptic Functions, 2nd ed, 1895, p. 380, Section 488.
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; Eq. (34.3).
R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, 1922, Vol. 2, see p. 375. Eqs. (17),(18),(19).
LINKS
T. D. Noe, Table of n, a(n) for n = 0..1000
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, p. 11.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Elliptic Lambda Function
FORMULA
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
Expansion of Fricke t(omega) = tau(omega) / 2 + 1 in powers of q = exp(2 Pi i omega).
Expansion of elliptic 1 / sqrt(1 - lambda(q)) = 1 / k'(q) in powers of the nome q = exp(Pi*i*z).
Euler transform of period 4 sequence [ 8, -4, 8, 0, ...]. - Michael Somos, Jul 07 2005
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
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.
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.
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
a(n) ~ exp(Pi*sqrt(2*n)) / (8 * 2^(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 28 2015
G.f.: exp(8*Sum_{k>=1} sigma(2*k - 1)*x^(2*k-1)/(2*k - 1)). - Ilya Gutkovskiy, Apr 19 2019
Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = (1/4)*sqrt(8 + 6*sqrt(2)). - Simon Plouffe, Mar 02 2021
From Peter Bala, Sep 25 2023: (Start)
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.
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)
EXAMPLE
G.f. = 1 + 8*q + 32*q^2 + 96*q^3 + 256*q^4 + 624*q^5 + 1408*q^6 + 3008*q^7 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ 1 / Sqrt[1 - InverseEllipticNomeQ @ q], {q, 0, n}]; (* Michael Somos, Aug 01 2011 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] / EllipticTheta[ 4, 0, q])^2, {q, 0, n}]; (* Michael Somos, Aug 01 2011 *)
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 *)
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 *)
PROG
(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 */
KEYWORD
nonn,nice
AUTHOR
STATUS
approved