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

%I A378807 #5 Dec 08 2024 02:44:12
%S A378807 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,
%T A378807 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,
%U A378807 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
%N A378807 Decimal expansion of Sum_{k>=1} (-1)^k/binomial(4*k, k) (negated).
%H A378807 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., Vol. 220 (2013), pp. 331-338.
%F A378807 Equals 4F3(1, 4/3, 5/3, 2; 5/4, 3/2, 7/4; -27/256) / 4, where 4F3 is a generalized hypergeometric function.
%F A378807 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).
%e A378807 -0.21833954717793443687099832102788539198304864029226...
%t A378807 RealDigits[HypergeometricPFQ[{1, 4/3, 5/3, 2}, {5/4, 3/2, 7/4}, -27/256]/ 4, 10, 120][[1]]
%Y A378807 Cf. A005810, A086465, A229703, A378806.
%K A378807 nonn,cons
%O A378807 0,1
%A A378807 _Amiram Eldar_, Dec 07 2024