

A257963


Decimal expansion of the integral_{x=0..1} arctan(arctanh(x))/x.


1



1, 0, 2, 5, 7, 6, 0, 5, 1, 0, 9, 3, 1, 3, 3, 0, 4, 5, 0, 3, 9, 8, 5, 4, 8, 6, 6, 0, 9, 6, 9, 5, 5, 2, 7, 9, 5, 3, 3, 4, 8, 7, 1, 8, 5, 6, 2, 1, 5, 0, 6, 9, 3, 9, 4, 2, 2, 3, 3, 8, 6, 8, 4, 4, 0, 1, 5, 8, 5, 1, 9, 2, 0, 8, 9, 9, 0, 7, 0, 9, 4, 2, 2, 2, 6, 7, 8, 7, 8, 7, 9, 1, 9, 7, 7, 9, 5, 3, 0, 7, 1, 3, 2, 9, 6
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OFFSET

1,3


COMMENTS

"The arctangent of the hyperbolic arctangent is analytic in the whole disk x < 1, and therefore, can be expanded into the MacLaurin series", see the first reference.


LINKS

Table of n, a(n) for n=1..105.
Iaroslav V. Blagouchine, Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to 1/pi, Journal of Mathematical Analysis and Applications (Elsevier), 2016. arXiv version, arXiv:1408.3902 [math.NT], 20142016.
Iaroslav V. Blagouchine, Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results, The Ramanujan Journal, Volume 35, Issue 1, pp. 21110, 2014, DOI: 10.1007/s1113901395285 PDF file


FORMULA

The integral is equivalent to Pi*(log(Gamma(1/Pi))  log(Gamma(1/2 + 1/Pi))  log(Pi)/2), see page 82 of the second reference.


EXAMPLE

= 1.02576051093133045039854866096955279533487185621506939422338684401585192089...


MAPLE

evalf(Pi*(log(GAMMA(1/Pi))  log(GAMMA(1/2 + 1/Pi))  log(Pi)/2), 120); # Vaclav Kotesovec, May 17 2015


MATHEMATICA

nn = 111; RealDigits[ NIntegrate[ ArcTan[ ArcTanh[ x]]/x, {x, 0, 1}, AccuracyGoal > nn, WorkingPrecision > nn], 10, nn][[1]] (* or *)
RealDigits[Pi (Log[Gamma[1/Pi]]  Log[Gamma[1/2 + 1/Pi]]  Log[Pi]/2), 10, 111][[1]] (* Robert G. Wilson v, May 14 2015 *)


CROSSREFS

Cf. A257957, A257955, A257958, A257959, A155968, A049541, A000796.
Sequence in context: A079378 A066035 A296168 * A167554 A202392 A290410
Adjacent sequences: A257960 A257961 A257962 * A257964 A257965 A257966


KEYWORD

nonn,cons


AUTHOR

Robert G. Wilson v, May 14 2015


STATUS

approved



