A004677 - OEIS

A004677

Numerator of 2^n*(3*n-3)!/( ((n-1)!)^3 * (2*n)! ).

1, 1, 1, 2, 11, 91, 17, 323, 4807, 3289, 8671, 11687, 15283, 10743949, 15189721, 21069613, 1339779509, 1339779509, 101007559, 101007559, 4215217889, 185371558793, 8059632991, 11489264051, 815737747621, 2203307656324321, 41571842572157, 3284175563200403

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

Pavel Valtr, The probability that n random points in a triangle are in convex position, Combinatorica, Vol. 16, No. 4 (1996), 567-573.

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

FORMULA

a(n)/A004824(n) ~ 3^(3*n-5/2) * exp(2*n) / (2^(n+2) * n^(2*n+3/2) * Pi^(3/2)). - Amiram Eldar, Oct 28 2025

EXAMPLE

Fractions begin: 1, 1, 1, 2/3, 11/36, 91/900, 17/675, 323/66150, 4807/6350400, 3289/34292160, 8671/857304000, 11687/12966723000, ...

MATHEMATICA

Table[Numerator[2^n*(3*n - 3)!/(((n - 1)!)^3*(2*n)!)], {n, 1, 50}] (* G. C. Greubel, Oct 12 2018 *)

PROG

(PARI) for(n=1, 50, print1(numerator(2^n*(3*n - 3)!/(((n - 1)!)^3*(2*n)!)), ", ")) \\ G. C. Greubel, Oct 12 2018

(Magma) [Numerator(2^n*Factorial(3*n - 3)/((Factorial(n - 1))^3*Factorial(2*n))): n in [1..50]]; // G. C. Greubel, Oct 12 2018

CROSSREFS

Cf. A000139, A004824 (denominators).

Sequence in context: A371537 A138552 A258221 * A266656 A094955 A352292

Adjacent sequences: A004674 A004675 A004676 * A004678 A004679 A004680

KEYWORD

nonn,frac,easy

AUTHOR

N. J. A. Sloane

STATUS

approved