%I #10 Nov 14 2020 05:10:19
%S 4,1,4,3,2,2,0,4,4,3,2,1,8,2,0,3,9,1,8,6,5,0,0,3,9,4,3,8,3,1,2,4,8,9,
%T 5,0,8,4,5,2,7,2,7,4,2,1,4,3,9,5,2,7,7,6,4,7,2,9,3,5,3,3,2,5,6,7,2,0,
%U 4,6,7,1,2,4,6,0,4,3,8,5,6,8,8,1,5,6,3,5,8,2,4,3,0,5,0,5,7,7,1,8,2,5,5,4,1
%N Decimal expansion of sum_{k>=1} 1/binomial(3k,k).
%H J. M. Borwein, R. Girgensohn, <a href="http://dx.doi.org/10.1007/s00010-005-2774-x">Evaluations of binomial series</a>, Aequat. Math. 70 (2005) 25-36, Eq. (39).
%e 0.4143220443218203918650039438312489508452...
%t HypergeometricPFQ[{1, 3/2, 2}, {4/3, 5/3}, 4/27]/3 // RealDigits[#, 10, 105]& // First (* _Jean-François Alcover_, Feb 18 2014 *)
%t Chop[N[(1/3174)*(552 + 2*(-110*69^(2/3)*(2/(-4761 + 997*Sqrt[69]))^(1/3) + 2^(2/3)*(69*(-4761 + 997*Sqrt[69]))^(1/3))* Log[997 + (1/3)*(26757728271/2 - (2973080919*Sqrt[69])/2)^(1/3) + (997*((1/2)*(9 + Sqrt[69]))^(1/3))/3^(2/3)] + (110*69^(2/3)*(1 - I*Sqrt[3])*(2/(-4761 + 997*Sqrt[69]))^(1/3) - 2^(2/3)*(1 + I*Sqrt[3])* (69*(-4761 + 997*Sqrt[69]))^(1/3))* Log[997 - (1/6)*(1 + I*Sqrt[3])*(26757728271/2 - (2973080919*Sqrt[69])/2)^(1/3) - (997*(1 - I*Sqrt[3])*((1/2)*(9 + Sqrt[69]))^(1/3))/(2*3^(2/3))] + (110*69^(2/3)*(1 + I*Sqrt[3])*(2/(-4761 + 997*Sqrt[69]))^(1/3) - 2^(2/3)*(1 - I*Sqrt[3])* (69*(-4761 + 997*Sqrt[69]))^(1/3))* Log[997 - (1/6)*(1 - I*Sqrt[3])*(26757728271/2 - (2973080919*Sqrt[69])/2)^(1/3) - (997*(1 + I*Sqrt[3])*((1/2)*(9 + Sqrt[69]))^(1/3))/(2*3^(2/3))]), 120]] (* _Vaclav Kotesovec_, Nov 14 2020 *)
%Y Cf. A073016.
%K nonn,cons
%O 0,1
%A _R. J. Mathar_, Sep 27 2013
%E More terms from _Jean-François Alcover_, Feb 18 2014