A004936 Numerator of (binomial(2*n-2,n-1)/n!)^2.

1, 1, 1, 25, 49, 49, 121, 20449, 20449, 5909761, 17631601, 17631601, 55190041, 55190041, 55190041, 46414824481, 154341336769, 154341336769, 427538329, 585299972401, 585299972401, 983889253606081, 3438962627443561, 3438962627443561, 7596668444022826249

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

a(n)/A005017(n) ~ 2^(4*n-5) * exp(2*n) / (n^(2*n+2) * Pi^2). - Amiram Eldar, Oct 28 2025

Example

Fractions begin: 1, 1, 1, 25/36, 49/144, 49/400, 121/3600, 20449/2822400, 20449/16257024, 5909761/32920473600, 17631601/823011840000, 17631601/8129341440000, ...

Mathematica

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

Prog

(PARI) a(n) = numerator((binomial(2*n-2, n-1)/n!)^2); \\ Michel Marcus, Jul 14 2022
(Magma) [Numerator((Catalan(n-1)/Factorial(n-1))^2): n in [1..40]]; // G. C. Greubel, Sep 12 2023
(SageMath) [numerator((catalan_number(n-1)/factorial(n-1))^2) for n in range(1, 41)] # G. C. Greubel, Sep 12 2023

Crossrefs

Cf. A000108, A005017 (denominators).

Sequence in context: A106632 A284666 A090093 * A062058 A273510 A390747

Adjacent sequences: A004933 A004934 A004935 * A004937 A004938 A004939

Keyword

nonn,frac

Author

N. J. A. Sloane

Status

approved