A395211 Numerator of the probability that the unique (n-1)-sphere passing through n+1 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.

2, 24, 286, 144000, 682341, 137682944000, 22222514956802, 10083790436267520000000, 1208420475776243722679, 6493940520603789372076706941108224, 20373929377943607470952397743467, 9231021478722028083971856549124150185073934598144, 292727737909189581001705180375682006378081965

Offset

2,1

Comments

The probability that the n+1 points lie on the same (n-1)-hyperplane and do not define an (n-1)-sphere is 0.

Links

Amiram Eldar, Table of n, a(n) for n = 2..56

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

Formula

Let f(n) = a(n)/A395212(n) if n is even, and a(n)*Pi^(n-1)/A395212(n) if n is odd. Then:

f(n) = 2^(n^2) * n^(n+1) * Pi^(n/2-1) * Gamma(n/2)^n * Gamma(n) * Gamma((n^2+1)/2)^2 / (Gamma((n+1)/2)^n * Gamma(n^2+n+1)).

f(n) ~ (2*Pi)^(n/2+1) / (exp(n+1/2-1/(6*n)) * n^((n+3)/2)).

Example

Fractions begin with 2/5, 24*Pi^2/1925, 286/8721, 144000*Pi^4/1784066921, 682341/394420000, 137682944000*Pi^6/373939020120748797, 22222514956802/326830530482421875, 10083790436267520000000*Pi^8/7747677348354274898150511341447, ...

Mathematica

a[n_] := Numerator[2^(n^2) * n^(n+1) * Pi^(n/2-1) * Gamma[n/2]^n * Gamma[n] * Gamma[(n^2+1)/2]^2 / (Gamma[(n+1)/2]^n * Gamma[n^2+n+1])] /. Pi -> 1; Array[a, 13, 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) = numerator(2^(n^2) * n^(n+1) * gamma_my(n/2)^n * gamma_my(n) * gamma_my((n^2+1)/2)^2 / (gamma_my((n+1)/2)^n * gamma_my(n^2+n+1)));

Crossrefs

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

Sequence in context: A230129 A355951 A065101 * A052739 A135389 A336310

Adjacent sequences: A395208 A395209 A395210 * A395212 A395213 A395214

Keyword

nonn,frac,easy

Author

Amiram Eldar, Apr 16 2026

Status

approved