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A203310 a(n) = A203309(n+1)/A203309(n).

%I A203310 #36 Aug 30 2023 02:01:59
%S A203310 1,2,15,252,7560,356400,24324300,2270268000,277880803200,
%T A203310 43197833952000,8315583035760000,1942008468966720000,
%U A203310 540988073497872000000,177227692877902867200000,67457290601651778828000000,29522484828017013792960000000,14721879100904484211422720000000
%N A203310 a(n) = A203309(n+1)/A203309(n).
%H A203310 G. C. Greubel, <a href="/A203310/b203310.txt">Table of n, a(n) for n = 0..200</a>
%F A203310 a(n) ~ sqrt(Pi) * 2^(n+3) * n^(2*n + 1/2) / exp(2*n). - _Vaclav Kotesovec_, Jan 25 2019
%F A203310 a(n) = (n!*(2*n+2)!)/(2^n*(n+2)!). - _G. C. Greubel_, Aug 29 2023
%p A203310 b:= proc(n) option remember; uses LinearAlgebra;
%p A203310       Determinant(VandermondeMatrix([i*(i+1)/2$i=1..n]))
%p A203310     end:
%p A203310 a:= n-> b(n+1)/b(n):
%p A203310 seq(a(n), n=0..16);  # _Alois P. Heinz_, Aug 29 2023
%t A203310 (* First program *)
%t A203310 f[j_]:= j*(j+1)/2; z = 15;
%t A203310 v[n_]:= Product[Product[f[k] - f[j], {j,k-1}], {k,2,n}]
%t A203310 Table[v[n], {n,z}]             (* A203309 *)
%t A203310 Table[v[n+1]/v[n], {n,0,z-1}]  (* A203310 *)
%t A203310 (* Second program *)
%t A203310 Table[(n!*(2*n+2)!)/(2^n*(n+2)!), {n,0,20}] (* _G. C. Greubel_, Aug 29 2023 *)
%o A203310 (Python)
%o A203310 from operator import mul
%o A203310 from functools import reduce
%o A203310 def f(n): return n*(n + 1)//2
%o A203310 def v(n): return 1 if n==1 else reduce(mul, (f(k) - f(j) for k in range(2, n + 1) for j in range(1, k)))
%o A203310 print([v(n + 1)//v(n) for n in range(1, 15)]) # _Indranil Ghosh_, Jul 24 2017
%o A203310 (Magma) F:= Factorial; [(F(n)*F(2*n+2))/(2^n*F(n+2)): n in [0..20]]; // _G. C. Greubel_, Aug 29 2023
%o A203310 (SageMath) f=factorial; [(f(n)*f(2*n+2))/(2^n*f(n+2)) for n in range(21)] # _G. C. Greubel_, Aug 29 2023
%Y A203310 Cf. A203306, A203309.
%K A203310 nonn
%O A203310 0,2
%A A203310 _Clark Kimberling_, Jan 01 2012
%E A203310 Name corrected by _Vaclav Kotesovec_, Jan 25 2019
%E A203310 a(0)=1 prepended by _Alois P. Heinz_, Aug 29 2023