A225847 Decimal expansion of Sum_{n>=1} 1/(n*binomial(4*n,n)).

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

Offset: 0

Author

Jean-François Alcover, May 17 2013

Keywords

• nonn
• cons

Examples

			0.269523929027742017317181647486329302840849825343266309815843772918628369827...
		

References

• George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), p. 60.

Links

• Necdet Batir and Anthony Sofo, On some series involving reciprocals of binomial coefficients, Appl. Math. Comp. 220 (2013) 331-338, Example 7.

Crossrefs

Cf. A073010, A210453, A378802.

Programs

Mathematica

(1/4)*HypergeometricPFQ[{1, 1, 4/3, 5/3}, {5/4, 3/2, 7/4}, 27/256] // RealDigits[#, 10, 105]& // First

Formula

Equals Integral_{x>0} ((3*x)/((1 + x)*(1 + 3*x + 6*x^2 + 4*x^3 + x^4))) dx.

Equals (3*c/(2*c^2+1)) * log((c-1)/(c+1)) + (3*(c-1)/(2*(2*c^2+1))) * sqrt(c/(c+2)) * arctan(2*sqrt(c^2+2*c)/(c^2+2*c-1)) + (3*(c+1)/(2*(2*c^2+1))) * sqrt(c/(c-2)) * arctan(2*sqrt(c^2-2*c)/(c^2-2*c-1)), where c = sqrt(1 + (16/sqrt(3)) * cos(arctan(sqrt(229/27))/3)) (Batir and Sofo, 2013). - Amiram Eldar, Dec 07 2024