%I #21 Apr 02 2020 04:31:22

%S 1,1,1,61,277,50521,41581,199360981,228135437,2404879675441,

%T 14814847529501,69348874393137901,238685140977801337,

%U 4087072509293123892361,454540704683713199807,441543893249023104553682821,2088463430347521052196056349

%N Numerators of Taylor series for log(tan(x)+1/cos(x)).

%C Absolute values of (reduced) numerators of Taylor series for the Gudermannian function gd(x)= 2*arctan(exp(x))-Pi/2. - Tamas Kalmar-Nagy (integers(AT)kalmarnagy.com), Sep 28 2007

%H Vincenzo Librandi, <a href="/A091912/b091912.txt">Table of n, a(n) for n = 0..100</a>

%H J. S. Robertson, <a href="http://www.jstor.org/stable/2687148">Gudermann and the Simple Pendulum</a>, The College Mathematics Journal, Vol. 28 (1997), No. 4, pp. 271-276.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Gudermannian.html">Gudermannian</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/InverseGudermannian.html">Inverse Gudermannian</a>

%F E.g.f.: sech x or gd x. - Tamas Kalmar-Nagy (integers(AT)kalmarnagy.com), Sep 28 2007

%e log(tan(x)+1/cos(x)) = x + 1/6*x^3 + 1/24*x^5 + 61/5040*x^7 + 277/72576*x^9 + ...

%e gd(x) = x - 1/6*x^3 + 1/24*x^5 - 61/5040*x^7 + 277/72576*x^9 + ....

%t Series[ArcTan[Sinh[x]], {x, 0, 30}] // CoefficientList[#, x]& // DeleteCases[#, 0]& // Numerator // Abs (* _Jean-François Alcover_, Feb 24 2014 *)

%t a[ n_] := (-1)^n Numerator @ SeriesCoefficient[ Gudermannian @ x, {x, 0, 2 n + 1}]; (* _Michael Somos_, Feb 24 2014 *)

%o (PARI) a(n)=local(X); if(n<0,0,X=x+O(x^(2*n+2)); numerator(polcoeff(log(tan(X)+1/cos(X)),2*n+1)))

%Y Cf. A000364, A028296.

%K nonn,frac,easy

%O 0,4

%A _Michael Somos_, Feb 12 2004

%E More terms from _Vincenzo Librandi_, Feb 26 2014