A004936 -id:A004936 - cp's OEIS frontend

A005017 Denominator of (binomial(2*n-2,n-1)/n!)^2.

1, 1, 1, 36, 144, 400, 3600, 2822400, 16257024, 32920473600, 823011840000, 8129341440000, 292656291840000, 3877578804363264, 58642395498086400, 844450495172444160000, 54044831691036426240000, 1161740555606493757440000, 76817130615613056614400

Offset: 1

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 1..270
Pavel Valtr, The probability that n random points in a triangle are in convex position, Combinatorica 16 (1996), no. 4, 567-573.
Eric Weisstein's World of Mathematics, Sylvester's Four-Point Problem.

Cf. A000108, A004936 (numerators).

[Denominator((Catalan(n-1)/Factorial(n-1))^2): n in [1..40]]; // G. C. Greubel, Sep 12 2023

Denominator[Table[(Binomial[2n-2,n-1]/n!)^2,{n,20}]] (* Harvey P. Dale, May 30 2012 *)

a(n) = denominator((binomial(2*n-2,n-1)/n!)^2); \\ Michel Marcus, Jul 14 2022

[denominator((catalan_number(n-1)/factorial(n-1))^2) for n in range(1,41)] # G. C. Greubel, Sep 12 2023

a(n) = denominator( (A000108(n-1)/(n-1)!)^2 ). - G. C. Greubel, Sep 12 2023

A242780 Decimal expansion of the maximum probability that the convex hull of four points, chosen at random inside a convex planar region, is a quadrilateral (Sylvester's four-point problem).

Original entry on oeis.org

7, 0, 4, 4, 7, 9, 8, 8, 1, 0, 4, 3, 1, 8, 1, 4, 9, 9, 9, 5, 5, 3, 5, 1, 5, 6, 5, 6, 3, 8, 2, 9, 4, 3, 8, 6, 5, 2, 8, 9, 5, 3, 5, 7, 3, 8, 7, 2, 6, 1, 4, 2, 3, 2, 5, 3, 3, 6, 4, 0, 3, 2, 3, 6, 4, 1, 9, 9, 5, 0, 6, 3, 8, 6, 0, 1, 4, 6, 6, 2, 9, 8, 5, 8, 9, 7, 2, 9, 5, 1, 0, 5, 0, 2, 6, 9, 6, 4, 0, 2, 9, 3, 6

Offset: 0

Jean-François Alcover, May 22 2014

It is proved that this maximum probability is achieved when the region is an ellipse (or a disk). [after Steven Finch]

			0.70447988104318149995535...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 8.18, p. 533.

Eric Weisstein's MathWorld, Sylvester's Four-Point Problem.

Cf. A004677, A004824, A004936, A005017, A051050, A051051.

RealDigits[1 - 35/(12*Pi^2), 10, 103] // First

1 - 35/(12*Pi^2) \\ Stefano Spezia, Dec 26 2024

Equals 1 - 35/(12*Pi^2).

cp's OEIS Frontend

A005017 Denominator of (binomial(2*n-2,n-1)/n!)^2.

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