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A203416 a(n) = A203415(n+1)/A203415(n).

%I A203416 #12 Feb 29 2024 01:46:23
%S A203416 3,10,56,120,432,12672,249600,873180,4838400,296110080,10786406400,
%T A203416 49621572000,355053404160,34613526528000,211189410432000,
%U A203416 1910897049600000,21311651380219200,274774815041126400,62908970812047360000
%N A203416 a(n) = A203415(n+1)/A203415(n).
%H A203416 G. C. Greubel, <a href="/A203416/b203416.txt">Table of n, a(n) for n = 1..350</a>
%t A203416 z=20;
%t A203416 nonprime = Join[{1}, Select[Range[250], CompositeQ]]; (* A018252 *)
%t A203416 f[j_]:= nonprime[[j]];
%t A203416 v[n_]:= Product[Product[f[k] - f[j], {j,1,k-1}], {k,2,n}];
%t A203416 d[n_]:= Product[(i-1)!, {i,1,n}];
%t A203416 Table[v[n], {n,1,z}]             (* A203415 *)
%t A203416 Table[v[n+1]/v[n], {n,1,z}]      (* this sequence *)
%t A203416 Table[v[n]/d[n], {n,1,z}]        (* A203417 *)
%o A203416 (Magma)
%o A203416 A018252:=[n : n in [1..250] | not IsPrime(n) ];
%o A203416 A203416:= func< n | n eq 1 select 3 else (&*[A018252[n+1] - A018252[j+1]: j in [0..n-1]]) >;
%o A203416 [A203416(n): n in [1..30]]; // _G. C. Greubel_, Feb 29 2024
%o A203416 (SageMath)
%o A203416 A018252=[n for n in (1..250) if not is_prime(n)]
%o A203416 def A203416(n): return product(A018252[n]-A018252[j] for j in range(n))
%o A203416 [A203416(n) for n in range(1,31)] # _G. C. Greubel_, Feb 29 2024
%Y A203416 Cf. A000040, A018252, A203415, A203417.
%K A203416 nonn
%O A203416 1,1
%A A203416 _Clark Kimberling_, Jan 01 2012