A225847 - OEIS

%I #14 Dec 08 2024 02:33:36

%S 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,

%T 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,

%U 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

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

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

%H Necdet Batir and Anthony Sofo, <a href="http://dx.doi.org/10.1016/j.amc.2013.05.053">On some series involving reciprocals of binomial coefficients</a>, Appl. Math. Comp. 220 (2013) 331-338, Example 7.

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

%F 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

%e 0.269523929027742017317181647486329302840849825343266309815843772918628369827...

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

%Y Cf. A073010, A210453, A378802.

%K nonn,cons

%O 0,1

%A _Jean-François Alcover_, May 17 2013