|
| |
|
|
A014969
|
|
Expansion of (theta_3 / theta_4)^2.
|
|
6
| |
|
|
1, 8, 32, 96, 256, 624, 1408, 3008, 6144, 12072, 22976, 42528, 76800, 135728, 235264, 400704, 671744, 1109904, 1809568, 2914272, 4640256, 7310592, 11404416, 17626944, 27009024, 41047992, 61905088, 92681664
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,2
|
|
|
COMMENTS
| Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
|
|
|
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. Eq. (17)
|
|
|
LINKS
| T. D. Noe, Table of n, a(n) for n=0..1000
M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
|
|
|
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 tau_8(omega) / 2 + 1 in powers of q = exp(2 pi i z).
Expansion of elliptic 1 / sqrt(1 - lambda(z)) = 1 / k' in powers of 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/theta_4)^2 = (Sum_{k} x^k^2)/(Sum_{k} (-x)^k^2)^2 = (Product_{k>0} (1-x^(4k-2))/((1-x^(4k-1))(1-x^(4k-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.
|
|
|
EXAMPLE
| 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 *)
|
|
|
PROG
| (PARI) {a(n) = local(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 */
|
|
|
CROSSREFS
| Cf. A107035, A131126, A139820.
Sequence in context: A159941 A053348 A019256 * A139820 A195590 A071345
Adjacent sequences: A014966 A014967 A014968 * A014970 A014971 A014972
|
|
|
KEYWORD
| nonn,nice
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
| |
|
|
