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Sum_{prime p <= n} T(n,p) = A359951(n). - Alois P. Heinz, Jan 20 2023
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b[n_, g_] := b[n, g] = If[n == 0, x^g, Sum[(j - 1)!*
b[n - j, GCD[g, j]] Binomial[n - 1, j - 1], {j, n}]];
T[n_] := CoefficientList[b[n, 0], x];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Aug 30 2021, after Alois P. Heinz *)
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Alois P. Heinz, <a href="/A346085/b346085.txt">Table of n, a(n) for Rows n = 0..10010140, flattened</a>
Alois P. Heinz, <a href="/A346085/b346085.txt">Table of n, a(n) for n = 0..10010</a>