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

%I A203308 #25 Aug 30 2023 02:12:37
%S A203308 1,1,20,9108,153675648,153926018668800,13624548214772203315200,
%T A203308 148312029363286484759480524800000,
%U A203308 262925014428462931164318003384701335633920000,96950311125839455466119755365478799838570665250861875200000
%N A203308 a(n) = A203306(n+1)/A203306(n).
%H A203308 G. C. Greubel, <a href="/A203308/b203308.txt">Table of n, a(n) for n = 0..29</a>
%F A203308 a(n) ~ (2*Pi)^(n/2) * n^(n*(2*n + 3)/2) / exp(n^2 - 13/12). - _Vaclav Kotesovec_, Jan 25 2019
%F A203308 a(n) = Product_{j=1..n} ((n+1)! - j!). - _G. C. Greubel_, Aug 30 2023
%t A203308 (* First program *)
%t A203308 f[j_]:= j!; z = 10;
%t A203308 v[n_]:= Product[Product[f[k] - f[j], {j,k-1}], {k,2,n}]
%t A203308 Table[v[n], {n,0,z}]          (* A203306 *)
%t A203308 Table[v[n+1]/v[n], {n,0,z}]   (* A203308 *)
%t A203308 (* Second program *)
%t A203308 Table[Product[(n+1)! - k!, {k,n}], {n,0,10}] (* _Vaclav Kotesovec_, Jan 25 2019 *)
%o A203308 (Python)
%o A203308 from sympy import factorial as f
%o A203308 from operator import mul
%o A203308 from functools import reduce
%o A203308 def v(n):
%o A203308     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)))
%o A203308 print([v(n + 1)//v(n) for n in range(16)]) # _Indranil Ghosh_, Jul 24 2017
%o A203308 (Magma) F:= Factorial; [1] cat [(&*[F(n+1) - F(j): j in [1..n]]): n in [1..20]]; // _G. C. Greubel_, Aug 30 2023
%o A203308 (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
%Y A203308 Cf. A203306, A323717.
%K A203308 nonn
%O A203308 0,3
%A A203308 _Clark Kimberling_, Jan 01 2012
%E A203308 a(0) = 1 prepended by _G. C. Greubel_, Aug 30 2023