A395208 - OEIS

%I #12 Apr 18 2026 06:16:22

%S 5,245,27,11011,20,138567,875,1834963,64827,153609433,679140,

%T 1090194825,226512,41812222997,200420649,945378466949,6213939875,

%U 7157865535471,1459710967,779542432563607,990190712160,9207723631443355,66209553337564,609237849980054157,174698555781400

%N Denominator of the probability that the unique circle passing through three points, selected independently and uniformly at random in an n-ball, is entirely contained within that n-ball.

%C The probability that the three points are collinear (i.e., lie on the same straight line) and do not define a circle is 0.

%H Amiram Eldar, <a href="/A395208/b395208.txt">Table of n, a(n) for n = 2..1001</a>

%H Fernando Affentranger, <a href="https://doi.org/10.1017/s0021900200027406">Random circles in the d-dimensional unit ball</a>, Journal of Applied Probability , Vol. 26, No. 2 (1989), p. 408-412; <a href="https://www.jstor.org/stable/3214047">JSTOR link</a>.

%e Fractions begin with 2/5, 12*Pi^2/245, 14/27, 600*Pi^2/11011, 11/20, 7840*Pi^2/138567, 494/875, 105840*Pi^2/1834963, 37145/64827, 8964648*Pi^2/153609433, 392863/679140, 64128064*Pi^2/1090194825, ...

%t a[n_] := Denominator[Pi * (n-1) * n^2 * Gamma[n/2]^3 * Gamma[(3*n-1)/2] / (6 * (2*n+1) * Gamma[(n+1)/2]^3 * Gamma[3*n/2])]; Array[a, 26, 2]

%o (PARI) gamma_my(n) = if(denominator(n) == 1, (n-1)!, my(m = n - 1/2); (2*m)! / (4^m * m!)); \\ gamma for integers, gamma(n)/sqrt(Pi) for half-integers

%o a(n) = denominator((n-1) * n^2 * gamma_my(n/2)^3 * gamma_my((3*n-1)/2) / (6 * (2*n+1) * gamma_my((n+1)/2)^3 * gamma_my(3*n/2)));

%Y Cf. A395207 (numerators), A395208, A395209, A395210, A395211, A395212, A395213, A395214.

%K nonn,frac,easy

%O 2,1

%A _Amiram Eldar_, Apr 16 2026