A335257 Numerators of expansion of arctanh(tan(x)) (odd powers only).

1, 2, 2, 244, 554, 202084, 166324, 1594887848, 456270874, 9619518701764, 59259390118004, 554790995145103208, 954740563911205348, 32696580074344991138888, 3636325637469705598456, 7064702291984369672858925136, 4176926860695042104392112698

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

Table of n, a(n) for n=1..17.

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 A350028 A262058

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