login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 56th year. In the past year we added 10000 new sequences and reached almost 9000 citations (which often say "discovered thanks to the OEIS").
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A091912 Numerators of Taylor series for log(tan(x)+1/cos(x)). 4
1, 1, 1, 61, 277, 50521, 41581, 199360981, 228135437, 2404879675441, 14814847529501, 69348874393137901, 238685140977801337, 4087072509293123892361, 454540704683713199807, 441543893249023104553682821, 2088463430347521052196056349 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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

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

FORMULA

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

EXAMPLE

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 + ....

MATHEMATICA

Series[ArcTan[Sinh[x]], {x, 0, 30}] // CoefficientList[#, x]& // DeleteCases[#, 0]& // Numerator // Abs (* Jean-Fran├žois Alcover, Feb 24 2014 *)

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

PROG

(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)))

CROSSREFS

Cf. A000364, A028296.

Sequence in context: A252803 A302282 A140854 * A142605 A142133 A302730

Adjacent sequences:  A091909 A091910 A091911 * A091913 A091914 A091915

KEYWORD

nonn,frac,easy

AUTHOR

Michael Somos, Feb 12 2004

EXTENSIONS

More terms from Vincenzo Librandi, Feb 26 2014

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 25 11:32 EST 2020. Contains 338623 sequences. (Running on oeis4.)