cp's OEIS Frontend

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

%I A203306 #28 Aug 30 2023 02:02:05
%S A203306 1,1,1,20,182160,27993556039680,4308936629569882673577984000,
%T A203306 58707314863972899718827044647532534690532556800000,
%U A203306 8707001005945253804913483804375384209011420702238388319242163029949808640000000000
%N A203306 Vandermonde determinant of (1!, 2!, 3!, ..., n!).
%C A203306 Each term divides its successor, as in A203308.
%H A203306 G. C. Greubel, <a href="/A203306/b203306.txt">Table of n, a(n) for n = 0..16</a>
%F A203306 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 A203306 with(LinearAlgebra):
%p A203306 a:= n-> Determinant(VandermondeMatrix([i!$i=1..n])):
%p A203306 seq(a(n), n=0..10);  # _Alois P. Heinz_, Jul 23 2017
%t A203306 f[j_]:= j!; z = 10;
%t A203306 v[n_]:= Product[Product[f[k] - f[j], {j,k-1}], {k,2,n}]
%t A203306 Table[v[n], {n,0,z}]        (* A203306 *)
%t A203306 Table[v[n+1]/v[n], {n,z}]   (* A203308 *)
%o A203306 (Python)
%o A203306 from sympy import factorial, prod
%o A203306 f = factorial
%o A203306 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 A203306 print([v(n) for n in range(11)]) # _Indranil Ghosh_, Jul 24 2017
%o A203306 (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 A203306 (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 A203306 Cf. A203308.
%K A203306 nonn
%O A203306 0,4
%A A203306 _Clark Kimberling_, Jan 01 2012
%E A203306 a(0)=1 prepended by _Alois P. Heinz_, Jul 23 2017
%E A203306 Offset corrected by _Vaclav Kotesovec_, Jan 25 2019