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

Author

Denis Roegel, May 28 2020

Keywords

• nonn

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).

Examples

			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 ...
		

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

• Lambert's use of the series
• Denis Roegel, Lambert's proof of the irrationality of Pi: Context and translation, hal-02984214 [math.HO], 2020.

Crossrefs

Cf. A335257.

Programs

Mathematica

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

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