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

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, 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, 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

Offset

0,1

Links

Table of n, a(n) for n=0..104.

Jonathan M. Borwein and Roland Girgensohn, Evaluations of binomial series, aequationes mathematicae, Vol. 70, No. 1 (2005), pp. 25-36. See p. 31, eq. (39).

Formula

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

Example

0.41432204432182039186500394383124895084527274214395..

Mathematica

HypergeometricPFQ[{1, 3/2, 2}, {4/3, 5/3}, 4/27]/3 // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Feb 18 2014 *)
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 *)

Crossrefs

Cf. A005809, A073016.

Sequence in context: A377120 A016686 A060037 * A321593 A173259 A021711

Adjacent sequences: A229702 A229703 A229704 * A229706 A229707 A229708

Keyword

nonn,cons

Author

R. J. Mathar, Sep 27 2013

Extensions

More terms from Jean-François Alcover, Feb 18 2014

Status

approved