A203308 a(n) = A203306(n+1)/A203306(n).

1, 1, 20, 9108, 153675648, 153926018668800, 13624548214772203315200, 148312029363286484759480524800000, 262925014428462931164318003384701335633920000, 96950311125839455466119755365478799838570665250861875200000

Offset: 0

Author

Clark Kimberling, Jan 01 2012

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..29

Crossrefs

Cf. A203306, A323717.

Programs

Magma

F:= Factorial; [1] cat [(&*[F(n+1) - F(j): j in [1..n]]): n in [1..20]]; // G. C. Greubel, Aug 30 2023

Mathematica

(* First program *)
f[j_]:= j!; z = 10;
v[n_]:= Product[Product[f[k] - f[j], {j,k-1}], {k,2,n}]
Table[v[n], {n,0,z}]          (* A203306 *)
Table[v[n+1]/v[n], {n,0,z}]   (* A203308 *)
(* Second program *)
Table[Product[(n+1)! - k!, {k,n}], {n,0,10}] (* Vaclav Kotesovec, Jan 25 2019 *)

Python

from sympy import factorial as f
from operator import mul
from functools import reduce
def v(n):
    return 1 if n<2 else reduce(mul, (f(k+1) - f(j) for k in range(1,n) for j in range(1, k+1)))
print([v(n + 1)//v(n) for n in range(16)]) # Indranil Ghosh, Jul 24 2017

SageMath

f=factorial; [product(f(n+1) - f(k) for k in range(1,n+1)) for n in range(21)] # G. C. Greubel, Aug 30 2023

Formula

a(n) ~ (2*Pi)^(n/2) * n^(n*(2*n + 3)/2) / exp(n^2 - 13/12). - Vaclav Kotesovec, Jan 25 2019

a(n) = Product_{j=1..n} ((n+1)! - j!). - G. C. Greubel, Aug 30 2023

Extensions

a(0) = 1 prepended by G. C. Greubel, Aug 30 2023