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

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A335257 Numerators of expansion of arctanh(tan(x)) (odd powers only). 1
 1, 2, 2, 244, 554, 202084, 166324, 1594887848, 456270874, 9619518701764, 59259390118004, 554790995145103208, 954740563911205348, 32696580074344991138888, 3636325637469705598456, 7064702291984369672858925136, 4176926860695042104392112698 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The numerators of a series used by Johann Heinrich Lambert (1728-1777) in expressing the relationship between a circular sector and a hyperbolic sector. LINKS Johann Heinrich Lambert, Mémoire sur quelques propriétés remarquables des quantités transcendentes circulaires et logarithmiques, Histoire de l'Académie Royale des Sciences et Belles-Lettres, 1761, volume XVII, Berlin, 1768, pp. 265-322. See also. Denis Roegel, Lambert's proof of the irrationality of Pi: Context and translation, hal-02984214 [math.HO], 2020. FORMULA a(n)/A335258(n) = A002436(n-1)/(2*n-1)!. - Andrew Howroyd, May 29 2020 EXAMPLE arctan(tanh(x)) = x - 2/3*x^3 + 2/3*x^5 - 244/315*x^7 + 554/567*x^9 ... arctanh(tan(x)) = x + 2/3*x^3 + 2/3*x^5 + 244/315*x^7 + 554/567*x^9 ... MATHEMATICA Numerator @ CoefficientList[Series[ArcTanh[Tan[x]], {x, 0, 34}], x][[2 ;; -1 ;; 2]] (* Amiram Eldar, May 30 2020 *) PROG (PARI) a(n)={numerator((-1)^(n-1)*(polcoef(atan(tanh(x + O(x^(2*n)))), 2*n-1)))} \\ Andrew Howroyd, May 29 2020 CROSSREFS Cf. A002436, A335258 (denominators). Sequence in context: A078241 A216812 A068103 * A119512 A262058 A067091 Adjacent sequences:  A335254 A335255 A335256 * A335258 A335259 A335260 KEYWORD nonn AUTHOR Denis Roegel, May 28 2020 EXTENSIONS Terms a(9) and beyond from Andrew Howroyd, May 29 2020 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.

Last modified July 31 13:25 EDT 2021. Contains 346373 sequences. (Running on oeis4.)