A051050 Numerator of the probability that the convex hull of n+2 randomly chosen points in the unit ball B^n has n+1 vertices.

1, 35, 9, 676039, 20000, 322476036831, 45956640625, 109701233401363445369, 3112866107675783424, 95848123849566788297513035239457, 43415602393213717594147325952, 3393946773664194957561187924967433315489530525, 2607758643018506362775246318628948148224

Offset

1,2

Links

Table of n, a(n) for n=1..13.

Christian Buchta, On a conjecture of R. E. Miles about the convex hull of random points, Monatshefte für Mathematik, Vol. 102 (1986), pp. 91-102.

H. Groemer, On some mean values associated with a randomly selected simplex in a convex set, Pacific Journal of Mathematics, Vol. 45, No. 2 (1973), pp. 525-533.

John F. C. Kingman, Random secants of a convex body, Journal of Applied Probability, Vol. 6, No. 3 (1969), pp. 660-672.

Norbert Peyerimhoff, Areas and intersections in convex domains, The American Mathematical Monthly, Vol. 104, No. 8 (1997), pp. 697-704.

Eric Weisstein's World of Mathematics, Sylvester's Four-Point Problem.

Formula

From Amiram Eldar, Oct 28 2025: (Start)

a(n) = numerator(((n+2) / 2^n) * binomial(n+1, (n+1)/2)^(n+1) / binomial((n+1)^2, (n+1)^2/2)).

a(n)/(A051051(n) * Pi^(n*(1-(n mod 2)))) ~ 1 / (exp(3/4 + 9/(32*n)) * n^(n/2-3/2) * (2*Pi)^(n/2)). (End)

Example

Probabilities begins: 1, 35/(12*Pi^2), 9/143, 676039/(648000*Pi^4), 20000/12964479, 322476036831/(1721036800000*Pi^6), 45956640625/2077805148460987, 109701233401363445369/(5041895218133760000000*Pi^8), ...

Mathematica

a[n_] := Numerator[(n+2) / 2^n * Binomial[n+1, (n+1)/2]^(n+1) / Binomial[(n+1)^2, (n+1)^2/2]]; Array[a, 13] (* Amiram Eldar, Oct 28 2025 *)

Crossrefs

Cf. A051051 (denominators).

Sequence in context: A034086 A277079 A248477 * A267198 A267140 A348893

Adjacent sequences: A051047 A051048 A051049 * A051051 A051052 A051053

Keyword

nonn,frac,easy

Author

Eric W. Weisstein

Extensions

Offset corrected by Amiram Eldar, Oct 28 2025

Status

approved