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