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Decimal expansion of (5 + 4*sqrt(5)*arcsch(2))/25.
5

%I #25 May 25 2023 03:10:02

%S 3,7,2,1,6,3,5,7,6,3,8,5,6,0,1,6,1,5,5,5,5,7,7,3,2,9,3,1,8,0,2,4,2,1,

%T 7,0,1,6,9,8,2,8,2,7,3,0,1,6,1,1,5,8,6,1,9,0,2,8,0,2,4,4,1,5,9,7,0,2,

%U 4,4,8,6,1,8,4,4,5,2,7,8,4,5,4,4,5,9,6,1,0,5,8,7,8,8,8,7,9,8,2

%N Decimal expansion of (5 + 4*sqrt(5)*arcsch(2))/25.

%H D. H. Lehmer, <a href="http://www.jstor.org/stable/2322496">Interesting series involving the Central Binomial Coefficient</a>, Am. Math. Monthly 92, No. 7 (1985) 449-457.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CentralBinomialCoefficient.html">Central Binomial Coefficient</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Inverse_hyperbolic_function#Series_expansions">Inverse hyperbolic function: Series expansions</a>.

%F Equals Sum_{n>=1} (-1)^(n-1)/binomial(2*n,n).

%e 0.37216357638560161555577...

%p 2/625*(14*sqrt(5)*log((1+sqrt(5))/2)+5) ; # _R. J. Mathar_, Mar 04 2009

%t RealDigits[(5 + 4*Sqrt[5]*ArcSinh[1/2])/25, 10, 120][[1]] (* _Amiram Eldar_, May 25 2023 *)

%o (PARI) suminf(n=1, (-1)^(n-1)/binomial(2*n,n)) \\ _Michel Marcus_, Jul 31 2015

%o (PARI) asinh(.5)*sqrt(5)*.16+.2 \\ Use \p99 to get 99 digits. - _M. F. Hasler_, Jul 31 2015

%Y Cf. A086466, A086467, A086468.

%K nonn,cons

%O 0,1

%A _Eric W. Weisstein_, Jul 21 2003

%E Corrected definition and digits by a factor of 25/24. - _R. J. Mathar_, Mar 04 2009