A395208 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.

5, 245, 27, 11011, 20, 138567, 875, 1834963, 64827, 153609433, 679140, 1090194825, 226512, 41812222997, 200420649, 945378466949, 6213939875, 7157865535471, 1459710967, 779542432563607, 990190712160, 9207723631443355, 66209553337564, 609237849980054157, 174698555781400

Offset

2,1

Comments

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

Links

Amiram Eldar, Table of n, a(n) for n = 2..1001

Fernando Affentranger, Random circles in the d-dimensional unit ball, Journal of Applied Probability , Vol. 26, No. 2 (1989), p. 408-412; JSTOR link.

Example

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, ...

Mathematica

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]

Prog

(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
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)));

Crossrefs

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

Sequence in context: A144999 A361538 A215910 * A097323 A166943 A125533

Adjacent sequences: A395205 A395206 A395207 * A395209 A395210 A395211

Keyword

nonn,frac,easy

Author

Amiram Eldar, Apr 16 2026

Status

approved