A051051 Denominators of the probability that the convex hull of n+2 randomly chosen points in the unit ball B^n has n+1 vertices (with factor of Pi^n dropped for n even).

1, 12, 143, 648000, 12964479, 1721036800000, 2077805148460987, 5041895218133760000000, 14154229032767142723739127, 51951524164830314976613655528865792, 26139242043667199795045152494924574325, 27693064436166084251915569647372450555221803794432

Offset

1,2

Links

Table of n, a(n) for n=1..12.

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

Formula

a(n) = denominator(((n+2) / 2^n) * binomial(n+1, (n+1)/2)^(n+1) / binomial((n+1)^2, (n+1)^2/2)) / Pi^(n*(1-(n mod 2))). - Amiram Eldar, Oct 28 2025

Mathematica

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

Crossrefs

Cf. A051050 (numerators).

Sequence in context: A172210 A171317 A004191 * A328468 A208382 A208070

Adjacent sequences: A051048 A051049 A051050 * A051052 A051053 A051054

Keyword

nonn,frac,easy

Author

Eric W. Weisstein

Extensions

Offset corrected by Amiram Eldar, Oct 28 2025

Status

approved