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MATHEMATICA
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f[j_] := j!; z = 8;
u[n_] := Product[f[j]^2 + f[k]^2, {j, 1, k - 1}]
v[n_] := Product[u[n], {k, 2, n}]
Table[v[n], {n, 1, z}] (* A203689 *)
Table[v[n + 1]/v[n], {n, 1, z}] (* A203690 *)
Table[Product[k!^2 + (n+1)!^2, {k, 1, n}], {n, 1, 10}] (* Vaclav Kotesovec, Nov 21 2023 *)
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