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

%I A335258 #21 Feb 24 2021 08:16:46
%S A335258 1,3,3,315,567,155925,93555,638512875,127702575,1856156927625,
%T A335258 7795859096025,49308808782358125,56894779364259375,
%U A335258 1298054391195577640625,95646113035463615625,122529844256906551386796875,47570410123269602303109375,2405873491984360136479756640625
%N A335258 Denominators of expansion of arctanh(tan(x)) (odd powers only).
%C A335258 The denominators of a series used by Johann Heinrich Lambert (1728-1777) in expressing the relationship between a circular sector and a hyperbolic sector.
%C A335258 Lambert gave a(1)-a(4).
%D A335258 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.
%H A335258 <a href="https://books.google.fr/books?id=sQIOAAAAQAAJ&amp;pg=PA319">Lambert's use of the series</a>
%H A335258 Denis Roegel, <a href="https://hal.archives-ouvertes.fr/hal-02984214">Lambert's proof of the irrationality of Pi: Context and translation</a>, hal-02984214 [math.HO], 2020.
%e A335258 arctan(tanh(x)) = x - 2/3*x^3 + 2/3*x^5 - 244/315*x^7 + 554/567*x^9 ...
%e A335258 arctanh(tan(x)) = x + 2/3*x^3 + 2/3*x^5 + 244/315*x^7 + 554/567*x^9 ...
%t A335258 Denominator @ CoefficientList[ Series[ArcTanh[Tan[x]], {x, 0, 36}], x][[2 ;; -1 ;; 2]] (* _Amiram Eldar_, Jun 04 2020 *)
%o A335258 (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
%Y A335258 Cf. A335257.
%K A335258 nonn
%O A335258 1,2
%A A335258 _Denis Roegel_, May 28 2020