A378807 Decimal expansion of Sum_{k>=1} (-1)^k/binomial(4*k, k) (negated).

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

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*d^2/((d^2-4)*(2*d^2+1)^2) + (3*d*(d^2-1)*(2*d^2-1)/(2*(2*d^2+1)^3)) * log(abs((d-1)/(d+1))) + (3*(d^2-1)*(2*d^4-2*d^3-7*d^2-3*d+1)/(4*d*(2*d^2+1)^3)) * (d/(d+2))^(3/2) * arctan(2*sqrt(d^2+2*d)/(d^2+2*d-1)) - (3*(d^2-1)*(2*d^4+2*d^3-7*d^2+3*d+1)/(4*d*(2*d^2+1)^3)) * (d/(d-2))^(3/2) * arctan(2*sqrt(d^2-2*d)/(d^2-2*d-1)), where d = sqrt(1 - (8/sqrt(3))*(((3*sqrt(3)+sqrt(283))/16)^(1/3) - (((3*sqrt(3)+sqrt(283))/16)^(-1/3)))) (Batir and Sofo, 2013, pp. 336-337, Example 4).

Example

-0.21833954717793443687099832102788539198304864029226...

Mathematica

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

Crossrefs

Cf. A005810, A086465, A229703, A378806.

Sequence in context: A176052 A351560 A140273 * A021462 A082834 A075647

Adjacent sequences: A378804 A378805 A378806 * A378808 A378809 A378810

Keyword

nonn,cons

Author

Amiram Eldar, Dec 07 2024

Status

approved