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
Keywords
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 ...
Links
- 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.
Programs
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Mathematica
Numerator @ CoefficientList[Series[ArcTanh[Tan[x]], {x, 0, 34}], x][[2 ;; -1 ;; 2]] (* Amiram Eldar, May 30 2020 *) -
PARI
a(n)={numerator((-1)^(n-1)*(polcoef(atan(tanh(x + O(x^(2*n)))), 2*n-1)))} \\ Andrew Howroyd, May 29 2020
Formula
Extensions
Terms a(9) and beyond from Andrew Howroyd, May 29 2020
Comments