A091912 Numerators of Taylor series for log(tan(x)+1/cos(x)).

1, 1, 1, 61, 277, 50521, 41581, 199360981, 228135437, 2404879675441, 14814847529501, 69348874393137901, 238685140977801337, 4087072509293123892361, 454540704683713199807, 441543893249023104553682821, 2088463430347521052196056349

Offset: 0

Author

Michael Somos, Feb 12 2004

Keywords

• nonn
• frac
• easy

Comments

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

Examples

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

References

• Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 33, equations 33:14:14 - 33:14:15 at page 318.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..100
• J. S. Robertson, Gudermann and the Simple Pendulum, The College Mathematics Journal, Vol. 28 (1997), No. 4, pp. 271-276.
• Eric Weisstein's World of Mathematics, Gudermannian.
• Eric Weisstein's World of Mathematics, Inverse Gudermannian.

Crossrefs

Cf. A000364, A028296.

Programs

Mathematica

Series[ArcTan[Sinh[x]], {x, 0, 30}] // CoefficientList[#, x]& // DeleteCases[#, 0]& // Numerator // Abs (* Jean-François Alcover, Feb 24 2014 *)
a[ n_] := (-1)^n Numerator @ SeriesCoefficient[ Gudermannian @ x, {x, 0, 2 n + 1}]; (* Michael Somos, Feb 24 2014 *)

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)))

Formula

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

Extensions

More terms from Vincenzo Librandi, Feb 26 2014