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%I #26 Feb 19 2021 19:49:37
%S 1,2,2,244,554,202084,166324,1594887848,456270874,9619518701764,
%T 59259390118004,554790995145103208,954740563911205348,
%U 32696580074344991138888,3636325637469705598456,7064702291984369672858925136,4176926860695042104392112698
%N Numerators of expansion of arctanh(tan(x)) (odd powers only).
%C The numerators of a series used by Johann Heinrich Lambert (1728-1777) in expressing the relationship between a circular sector and a hyperbolic sector.
%H Johann Heinrich Lambert, <a href="http://www.kuttaka.org/~JHL/L1768b.html">Mémoire sur quelques propriétés remarquables des quantités transcendentes circulaires et logarithmiques</a>, Histoire de l'Académie Royale des Sciences et Belles-Lettres, 1761, volume XVII, Berlin, 1768, pp. 265-322. See <a href="https://books.google.fr/books?id=sQIOAAAAQAAJ&pg=PA319">also</a>.
%H 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.
%F a(n)/A335258(n) = A002436(n-1)/(2*n-1)!. - _Andrew Howroyd_, May 29 2020
%e arctan(tanh(x)) = x - 2/3*x^3 + 2/3*x^5 - 244/315*x^7 + 554/567*x^9 ...
%e arctanh(tan(x)) = x + 2/3*x^3 + 2/3*x^5 + 244/315*x^7 + 554/567*x^9 ...
%t Numerator @ CoefficientList[Series[ArcTanh[Tan[x]], {x, 0, 34}], x][[2 ;; -1 ;; 2]] (* _Amiram Eldar_, May 30 2020 *)
%o (PARI) a(n)={numerator((-1)^(n-1)*(polcoef(atan(tanh(x + O(x^(2*n)))), 2*n-1)))} \\ _Andrew Howroyd_, May 29 2020
%Y Cf. A002436, A335258 (denominators).
%K nonn
%O 1,2
%A _Denis Roegel_, May 28 2020
%E Terms a(9) and beyond from _Andrew Howroyd_, May 29 2020
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