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A335257 Numerators of expansion of arctanh(tan(x)) (odd powers only). 1
1, 2, 2, 244, 554, 202084, 166324, 1594887848, 456270874, 9619518701764, 59259390118004, 554790995145103208, 954740563911205348, 32696580074344991138888, 3636325637469705598456, 7064702291984369672858925136, 4176926860695042104392112698 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

LINKS

Table of n, a(n) for n=1..17.

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.

FORMULA

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

EXAMPLE

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 ...

MATHEMATICA

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

PROG

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

CROSSREFS

Cf. A002436, A335258 (denominators).

Sequence in context: A078241 A216812 A068103 * A119512 A262058 A067091

Adjacent sequences:  A335254 A335255 A335256 * A335258 A335259 A335260

KEYWORD

nonn

AUTHOR

Denis Roegel, May 28 2020

EXTENSIONS

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

STATUS

approved

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Last modified July 31 13:25 EDT 2021. Contains 346373 sequences. (Running on oeis4.)