A000819 E.g.f.: cos(x)^2 / cos(2x) = Sum_{n >= 0} a(n) * x^(2n) / (2n)!.

1, 2, 40, 1952, 177280, 25866752, 5535262720, 1633165156352, 635421069967360, 315212388819402752, 194181169538675507200, 145435130631317935357952, 130145345400688287667978240, 137139396592145493713802493952

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Examples

			G.f. = 1 + 2*x + 40*x^2 + 1952*x^3 + 177280*x^4 + 25866752*x^5 + ... - _Michael Somos_, Apr 04 2017
		

Links

• R. J. Mathar, Table of n, a(n) for n = 0..200

Crossrefs

Essentially the same as A000816.

Second column of array A103905.

Cf. A000364, A002436.

Programs

Mathematica

With[{nn=30},Take[CoefficientList[Series[Cos[x]^2/Cos[2x],{x,0,nn}],x] Range[ 0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Jul 06 2014 *)
a[ n_] := If[ n < 0, 0, With[ {m = 2 n}, m! SeriesCoefficient[ 1 / (1 - Tan[x]^2), {x, 0, m}]]]; (* Michael Somos, Apr 04 2017 *)

PARI

{a(n) = my(m); if( n<0, 0, m = 2*n; m! * polcoeff( 1 / (1 - tan(x + x * O(x^m))^2), m))}; /* Michael Somos, Apr 04 2017 */

Formula

E.g.f.: cos(x)^2/cos(2x)=1/Q(0)+1/2; Q(k)=1+1/(1-2*(x^2)/(2*(x^2)-(k+1)*(2k+1)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 18 2011

From Michael Somos, Apr 04 2017: (Start)

E.g.f.: cos(x)^2 / cos(2*x) = (1 + sec(2*x)) / 2 = tan(2*x) / (2 * tan(x)) = 1 / (1 - tan(x)^2).

a(n) = A000816(n) unless n=0.

a(n) = 1/2 * A002436(n) unless n=0.

a(n) = 2^(2*n - 1) * A000364(n). (End)