A395207 - OEIS

%I #20 Apr 26 2026 22:36:39

%S 2,12,14,600,11,7840,494,105840,37145,8964648,392863,64128064,131905,

%T 2473511040,117285338,56166368544,3650109911,426684744200,860010277,

%U 46593974066640,584803025179,551568386713344,39181802686993,36562298361680000,103559043001879,3153558501219672000

%N Numerator 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, divided by Pi^2 if n is odd.

%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="/A395207/b395207.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>.

%F Let f(n) = a(n)/A395208(n) if n is even, and a(n)*Pi^2/A395208(n) if n is odd. Then:

%F f(n) = 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)).

%F Limit_{n->oo} f(n) = Pi/(3*sqrt(3)) = 0.604599... (A073010).

%F f(n) ~ Pi/(3*sqrt(3))*(1 - 1/(2*n) - 7/(36*n^2) - 25/(72*n^3) + 11/(432*n^4) + ...).

%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_] := Numerator[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])] /. Pi -> 1; 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) = numerator((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. A002388, A073010, A395208 (denominators), A395209, A395210, A395211, A395212, A395213, A395214.

%K nonn,frac,easy

%O 2,1

%A _Amiram Eldar_, Apr 16 2026