A395209 Numerator of the probability that the unique sphere passing through four 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.

24, 11, 50, 221, 980000, 7429, 92610, 56695, 4390848, 112375, 529056528, 156835045, 50530296960, 622368971, 199454433750, 146078888479, 188594656936400, 201325631619, 3404944258716, 340001758825, 1988989030875392, 17596887808247, 36469027192390000, 60374922070095457

Offset

3,1

Comments

The probability that the four points are coplanar (i.e., lie on the same plane) and do not define a sphere is 0.

Links

Amiram Eldar, Table of n, a(n) for n = 3..1002

Fernando Affentranger, Random spheres in a convex body, Archiv der Mathematik, Vol. 55 (1990), pp. 74-81.

Formula

Let f(n) = a(n)/A395210(n) if n is even, and a(n)*Pi^2/A395210(n) if n is odd. Then:

f(n) = 4 * n^3 * Pi * Gamma(2*n-1) * Gamma((3*n+5)/2) * Gamma(n/2)^4 / (9 * binomial(3*n+3, 3) * Gamma(3*n/2) * Gamma((n-2)/2) * Gamma(2*n+1) * Gamma((n+1)/2)^3).

Limit_{n->oo} f(n) = Pi/(6*sqrt(3)) = 0.302299... (A381671).

f(n) ~ Pi/(6*sqrt(3))*(1 - 3/(2*n) - 19/(36*n^2) - 67/(72*n^3) + 23/(432*n^4) + ...).

Example

Fractions begin with 24*Pi^2/1925, 11/63, 50*Pi^2/2431, 221/1000, 980000*Pi^2/41431533, 7429/30625, 92610*Pi^2/3662497, 56695/222264, 4390848*Pi^2/166966775, 112375/426888, 529056528*Pi^2/19604380625, 156835045/583041888, ...

Mathematica

a[n_] := Numerator[4 * n^3 * Pi * Gamma[2*n-1] * Gamma[(3*n+5)/2] * Gamma[n/2]^4 / (9 * Binomial[3*n+3, 3] * Gamma[3*n/2] * Gamma[(n-2)/2] * Gamma[2*n+1] * Gamma[(n+1)/2]^3)] /. Pi -> 1; Array[a, 24, 3]

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) = numerator(4 * n^3 * gamma_my(2*n-1) * gamma_my((3*n+5)/2) * gamma_my(n/2)^4 / (9 * binomial(3*n+3, 3) * gamma_my(3*n/2) * gamma_my((n-2)/2) * gamma_my(2*n+1) * gamma_my((n+1)/2)^3));

Crossrefs

Cf. A002388, A381671, A395210 (denominators), A395207, A395208, A395211, A395212, A395213, A395214.

Sequence in context: A241441 A343389 A267024 * A259337 A118983 A033968

Adjacent sequences: A395206 A395207 A395208 * A395210 A395211 A395212

Keyword

nonn,frac,easy

Author

Amiram Eldar, Apr 16 2026

Status

approved