A009764 Expansion of e.g.f. tan(x)^2 (even powers only).
0, 2, 16, 272, 7936, 353792, 22368256, 1903757312, 209865342976, 29088885112832, 4951498053124096, 1015423886506852352, 246921480190207983616, 70251601603943959887872, 23119184187809597841473536, 8713962757125169296170811392, 3729407703720529571097509625856
Offset: 0
Examples
(tan x)^2 = x^2 + 2/3*x^4 + 17/45*x^6 + 62/315*x^8 + ...
Programs
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Mathematica
With[{nn=30},Take[CoefficientList[Series[Tan[x]^2,{x,0,nn}],x] Range[0,nn]!, {1,-1,2}]] (* Harvey P. Dale, Oct 04 2011 *)
Formula
a(n) = (2*n)! * [x^(2*n)] tan(x)^2.
(tan(z))^2 = z^2/(1-z^2)*( 1 +2*z^2/( (z^2-1)*G(0)-2*z^2)), G(k) = (k+2)*(2*k+3)-2*z^2+2*z^2*(k+2)*(2*k+3)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 15 2011
(tan(z))^2 = z^2/(G(0)+z^2) where G(k) = (k+1)*(2*k+1)-2*z^2+2*z^2*(k+1)*(2*k+1)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 15 2011
G.f.: -1 + 1/G(0) where G(k) = 1 - (k+1)*(k+2)*x/G(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Aug 10 2012
G.f.: 1/G(0)-1 where G(k) = 1 - 2*x*(2*k+1)^2 - x^2*(2*k+1)*(2*k+2)^2*(2*k+3)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 13 2013
G.f.: (1/G(0)-1)*sqrt(-x), where G(k) = 1 - sqrt(-x) - x*(k+1)^2/G(k+1); (continued fraction). - Sergei N. Gladkovskii, May 29 2013
G.f.: Q(0)-1, where Q(k) = 1 - x*(k+1)*(k+2)/( x*(k+1)*(k+2) - 1/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 14 2013
Extensions
Extended and signs tested by Olivier Gérard, Mar 15 1997
More terms from Harvey P. Dale, Oct 04 2011