A203306 - OEIS

%I #28 Aug 30 2023 02:02:05

%S 1,1,1,20,182160,27993556039680,4308936629569882673577984000,

%T 58707314863972899718827044647532534690532556800000,

%U 8707001005945253804913483804375384209011420702238388319242163029949808640000000000

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

%C Each term divides its successor, as in A203308.

%H G. C. Greubel, <a href="/A203306/b203306.txt">Table of n, a(n) for n = 0..16</a>

%F 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

%p with(LinearAlgebra):

%p a:= n-> Determinant(VandermondeMatrix([i!$i=1..n])):

%p seq(a(n), n=0..10);  # _Alois P. Heinz_, Jul 23 2017

%t f[j_]:= j!; z = 10;

%t v[n_]:= Product[Product[f[k] - f[j], {j,k-1}], {k,2,n}]

%t Table[v[n], {n,0,z}]        (* A203306 *)

%t Table[v[n+1]/v[n], {n,z}]   (* A203308 *)

%o (Python)

%o from sympy import factorial, prod

%o f = factorial

%o 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))

%o print([v(n) for n in range(11)]) # _Indranil Ghosh_, Jul 24 2017

%o (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

%o (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

%Y Cf. A203308.

%K nonn

%O 0,4

%A _Clark Kimberling_, Jan 01 2012

%E a(0)=1 prepended by _Alois P. Heinz_, Jul 23 2017

%E Offset corrected by _Vaclav Kotesovec_, Jan 25 2019