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A229705 Decimal expansion of Sum_{k>=1} 1/binomial(3k,k).

%I A229705 #19 Dec 07 2024 05:40:42
%S A229705 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 A229705 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 A229705 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 A229705 Decimal expansion of Sum_{k>=1} 1/binomial(3k,k).
%H A229705 Jonathan M. Borwein and Roland Girgensohn, <a href="https://doi.org/10.1007/s00010-005-2774-x">Evaluations of binomial series</a>, aequationes mathematicae, Vol. 70, No. 1 (2005), pp. 25-36. See p. 31, eq. (39).
%F A229705 Equals 4/23 + (2/23) * Sum_{r: 23*r^3 + 55*r + 23 = 0} r * log(1987 - 598*r + 621*r^2) (Borwein and Girgensohn, 2005). - _Amiram Eldar_, Dec 07 2024
%e A229705 0.41432204432182039186500394383124895084527274214395..
%t A229705 HypergeometricPFQ[{1, 3/2, 2}, {4/3, 5/3}, 4/27]/3 // RealDigits[#, 10, 105]& // First (* _Jean-François Alcover_, Feb 18 2014 *)
%t A229705 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 A229705 Cf. A005809, A073016.
%K A229705 nonn,cons
%O A229705 0,1
%A A229705 _R. J. Mathar_, Sep 27 2013
%E A229705 More terms from _Jean-François Alcover_, Feb 18 2014