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

%I A000819 #31 Dec 30 2017 14:33:34
%S A000819 1,2,40,1952,177280,25866752,5535262720,1633165156352,635421069967360,
%T A000819 315212388819402752,194181169538675507200,145435130631317935357952,
%U A000819 130145345400688287667978240,137139396592145493713802493952
%N A000819 E.g.f.: cos(x)^2 / cos(2x) = Sum_{n >= 0} a(n) * x^(2n) / (2n)!.
%H A000819 R. J. Mathar, <a href="/A000819/b000819.txt">Table of n, a(n) for n = 0..200</a>
%F A000819 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
%F A000819 From _Michael Somos_, Apr 04 2017: (Start)
%F A000819 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).
%F A000819 a(n) = A000816(n) unless n=0.
%F A000819 a(n) = 1/2 * A002436(n) unless n=0.
%F A000819 a(n) = 2^(2*n - 1) * A000364(n). (End)
%e A000819 G.f. = 1 + 2*x + 40*x^2 + 1952*x^3 + 177280*x^4 + 25866752*x^5 + ... - _Michael Somos_, Apr 04 2017
%t A000819 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 *)
%t A000819 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 *)
%o A000819 (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 */
%Y A000819 Essentially the same as A000816.
%Y A000819 Second column of array A103905.
%Y A000819 Cf. A000364, A002436.
%K A000819 nonn
%O A000819 0,2
%A A000819 _N. J. A. Sloane_