A276483 Decimal expansion of Sum_{k>=0} (2*k+1)/binomial(4*k,2*k).

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

Offset: 1

Author

Ilya Gutkovskiy, Sep 05 2016

Keywords

• nonn
• cons

Examples

			1.57976837955402077524299785912344486...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Eric Weisstein's MathWorld, Catalan Number

Crossrefs

Cf. A000108, A001622, A048990, A121839, A268813.

Programs

Magma

SetDefaultRealField(RealField(100)); R:=RealField(); 2*Pi(R)/(9*Sqrt(3)) - 4*(3*Sqrt(5)*Log((1+Sqrt(5))/2) - 40)/125; // G. C. Greubel, Nov 04 2018

Mathematica

RealDigits[2 (Pi/(9 Sqrt[3])) - 4 ((3 Sqrt[5] Log[GoldenRatio] - 40)/125), 10, 120][[1]]
RealDigits[HypergeometricPFQ[{1, 1, 3/2}, {1/4, 3/4}, 1/16], 10, 120][[1]]

PARI

suminf(k=0, 1/(binomial(4*k,2*k)/(2*k+1))) \\ Michel Marcus, Sep 06 2016

PARI

default(realprecision, 100); 2*Pi/(9*sqrt(3)) - 4*(3*sqrt(5)*log((1+sqrt(5))/2) - 40)/125 \\ G. C. Greubel, Nov 04 2018

Formula

Equals 2*Pi/(9*sqrt(3)) - 4*(3*sqrt(5)*log(phi) - 40)/125, where phi is the golden ratio (A001622).

Equals Sum_{k>=0} 1/Catalan number(2k).

Equals Sum_{k>=0} 1/A000108(2k).

Equals Sum_{k>=0} 1/A048990(k).