A378806 Decimal expansion of Sum_{k>=1} 1/binomial(4*k, k).

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

Offset

0,1

Links

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

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

Formula

Equals 4F3(1, 4/3, 5/3, 2; 5/4, 3/2, 7/4; 27/256) / 4, where 4F3 is a generalized hypergeometric function.

Equals 27*c^2/((c^2-4)*(2*c^2+1)^2) + (3*c*(c^2-1)*(2*c^2-1)/(2*(2*c^2+1)^3)) * log((c-1)/(c+1)) + (3*(c^2-1)*(2*c^4-2*c^3-7*c^2-3*c+1)/(4*c*(2*c^2+1)^3)) * (c/(c+2))^(3/2) * arctan(2*sqrt(c^2+2*c)/(c^2+2*c-1)) + (3*(c^2-1)*(2*c^4+2*c^3-7*c^2+3*c+1)/(4*c*(2*c^2+1)^3)) * (c/(c-2))^(3/2) * 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, p. 337, Example 9).

Example

0.29088207152128721276259725668681035773368176167609...

Mathematica

RealDigits[HypergeometricPFQ[{1, 4/3, 5/3, 2}, {5/4, 3/2, 7/4}, 27/256] / 4, 10, 120][[1]]

Crossrefs

Cf. A005810, A073016, A229705, A378807.

Sequence in context: A021348 A346833 A192938 * A188619 A020817 A245217

Adjacent sequences: A378803 A378804 A378805 * A378807 A378808 A378809

Keyword

nonn,cons

Author

Amiram Eldar, Dec 07 2024

Status

approved