A203306 Vandermonde determinant of (1!, 2!, 3!, ..., n!).

1, 1, 1, 20, 182160, 27993556039680, 4308936629569882673577984000, 58707314863972899718827044647532534690532556800000, 8707001005945253804913483804375384209011420702238388319242163029949808640000000000

Offset: 0

Author

Clark Kimberling, Jan 01 2012

Keywords

• nonn

Comments

Each term divides its successor, as in A203308.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..16

Crossrefs

Cf. A203308.

Programs

Magma

F:= Factorial; [1,1] cat [(&*[(&*[F(k+1) - F(j): j in [1..k]]): k in [1..n-1]]): n in [2..20]]; // G. C. Greubel, Aug 30 2023

Maple

with(LinearAlgebra):
a:= n-> Determinant(VandermondeMatrix([i!$i=1..n])):
seq(a(n), n=0..10);  # Alois P. Heinz, Jul 23 2017

Mathematica

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,z}]   (* A203308 *)

Python

from sympy import factorial, prod
f = factorial
def v(n): return 1 if n<2 else prod(f(k) - f(j) for k in range(2, n + 1) for j in range(1, k))
print([v(n) for n in range(11)]) # Indranil Ghosh, Jul 24 2017

SageMath

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

Formula

a(n) ~ c * (2*Pi)^(n*(n-1)/4) * n^(n^3/3 + n^2/4 - 7*n/12 - 11/8) / exp(4*n^3/9 - n^2/8 - n), where c = A323720 = 0.29363504888070220142364974947015983077985979... - Vaclav Kotesovec, Jan 25 2019

Extensions

a(0)=1 prepended by Alois P. Heinz, Jul 23 2017

Offset corrected by Vaclav Kotesovec, Jan 25 2019