A001459 a(n) = (5*n)!/((2*n)!*(2*n)!*n!).

1, 30, 3150, 420420, 62355150, 9816086280, 1605660228900, 269764879032000, 46225898052627150, 8042050347997165500, 1415997888807961859400, 251762943910387780962000, 45125969443194371927422500, 8143514687130622653091029120, 1478138194032735032800001630400

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Links

• James Spahlinger, Table of n, a(n) for n = 0..400
• Romeo Meštrović, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.

Crossrefs

Cf. A006480.

Programs

Maple

f := n->(5*n)!/((2*n)!*(2*n)!*n!);

Mathematica

Table[(5 n)!/((2 n)! (2 n)!*n!), {n, 0, 12}] (* Michael De Vlieger, Feb 07 2017 *)

Formula

From Ilya Gutkovskiy, Feb 07 2017: (Start)

O.g.f.: 4F3(1/5,2/5,3/5,4/5; 1/2,1/2,1; 3125*x/16).

E.g.f.: 4F4(1/5,2/5,3/5,4/5; 1/2,1/2,1,1; 3125*x/16).

a(n) ~ 5^(5*n+1/2)/(4*Pi*16^n*n). (End)

From Peter Bala, Sep 20 2021: (Start)

a(n) = 5*(5*n - 1)*(5*n - 2)*(5*n - 3)*(5*n - 4)/(4*n^2*(2*n - 1)^2)*a(n-1).

a(n) = Sum_{k = n..3*n} binomial(3*n,k)^2*binomial(k,n). Cf. A006480.

Congruences: a(n*p^k) == a(n*p^(k-1)) ( mod p^(3*k) ) for any prime p >= 5 and any positive integers n and k (write a(n) as C(5*n,2*n)*C(3*n,n) and apply Mestrovic, equation 39, p. 12). (End)