A335258 Denominators of expansion of arctanh(tan(x)) (odd powers only).

1, 3, 3, 315, 567, 155925, 93555, 638512875, 127702575, 1856156927625, 7795859096025, 49308808782358125, 56894779364259375, 1298054391195577640625, 95646113035463615625, 122529844256906551386796875, 47570410123269602303109375, 2405873491984360136479756640625

Offset

1,2

Comments

The denominators of a series used by Johann Heinrich Lambert (1728-1777) in expressing the relationship between a circular sector and a hyperbolic sector.

Lambert gave a(1)-a(4).

References

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.

Links

Table of n, a(n) for n=1..18.

Lambert's use of the series

Denis Roegel, Lambert's proof of the irrationality of Pi: Context and translation, hal-02984214 [math.HO], 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

Denominator @ CoefficientList[ Series[ArcTanh[Tan[x]], {x, 0, 36}], x][[2 ;; -1 ;; 2]] (* Amiram Eldar, Jun 04 2020 *)

Prog

(PARI) my(x='x+O('x^40), v=Vec(atanh(tan(x)))); apply(denominator, vector(#v\2, k, v[2*k-1])) \\ Michel Marcus, Jun 05 2020

Crossrefs

Cf. A335257.

Sequence in context: A007301 A009715 A344283 * A292163 A242886 A221947

Adjacent sequences: A335255 A335256 A335257 * A335259 A335260 A335261

Keyword

nonn

Author

Denis Roegel, May 28 2020

Status

approved