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a(n) = A203415(n+1)/A203415(n).
4

%I #12 Feb 29 2024 01:46:23

%S 3,10,56,120,432,12672,249600,873180,4838400,296110080,10786406400,

%T 49621572000,355053404160,34613526528000,211189410432000,

%U 1910897049600000,21311651380219200,274774815041126400,62908970812047360000

%N a(n) = A203415(n+1)/A203415(n).

%H G. C. Greubel, <a href="/A203416/b203416.txt">Table of n, a(n) for n = 1..350</a>

%t z=20;

%t nonprime = Join[{1}, Select[Range[250], CompositeQ]]; (* A018252 *)

%t f[j_]:= nonprime[[j]];

%t v[n_]:= Product[Product[f[k] - f[j], {j,1,k-1}], {k,2,n}];

%t d[n_]:= Product[(i-1)!, {i,1,n}];

%t Table[v[n], {n,1,z}] (* A203415 *)

%t Table[v[n+1]/v[n], {n,1,z}] (* this sequence *)

%t Table[v[n]/d[n], {n,1,z}] (* A203417 *)

%o (Magma)

%o A018252:=[n : n in [1..250] | not IsPrime(n) ];

%o A203416:= func< n | n eq 1 select 3 else (&*[A018252[n+1] - A018252[j+1]: j in [0..n-1]]) >;

%o [A203416(n): n in [1..30]]; // _G. C. Greubel_, Feb 29 2024

%o (SageMath)

%o A018252=[n for n in (1..250) if not is_prime(n)]

%o def A203416(n): return product(A018252[n]-A018252[j] for j in range(n))

%o [A203416(n) for n in range(1,31)] # _G. C. Greubel_, Feb 29 2024

%Y Cf. A000040, A018252, A203415, A203417.

%K nonn

%O 1,1

%A _Clark Kimberling_, Jan 01 2012