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User: Denis Roegel

Denis Roegel has authored 2 sequences.

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

Denis Roegel, May 28 2020

nonn

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

			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 ...

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.

Cf. A002436, A335258 (denominators).

Numerator @ CoefficientList[Series[ArcTanh[Tan[x]], {x, 0, 34}], x][[2 ;; -1 ;; 2]] (* Amiram Eldar, May 30 2020 *)

a(n)={numerator((-1)^(n-1)*(polcoef(atan(tanh(x + O(x^(2*n)))), 2*n-1)))} \\ Andrew Howroyd, May 29 2020

a(n)/A335258(n) = A002436(n-1)/(2*n-1)!. - Andrew Howroyd, May 29 2020

Terms a(9) and beyond from Andrew Howroyd, May 29 2020

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

Original entry on oeis.org

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

Offset: 1

Denis Roegel, May 28 2020

nonn

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).

			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 ...

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.

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

Cf. A335257.

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

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

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User: Denis Roegel

A335257 Numerators of expansion of arctanh(tan(x)) (odd powers only).

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