Clear[nn, t, n, k, i, s];
nn = 12;
t[n_, 1] = 1;
t[1, k_] = 1;
t[n_, k_] :=
  t[n, k] =
   If[n >= k, -Sum[t[n - i, k], {i, 1, k - 1}], -Sum[
      t[k - i, n], {i, 1, n - 1}]];
Exp[Table[
  Limit[Zeta[s]*Sum[If[n == 1, 0, t[n, k]]/k^(s - 1), {k, 1, n}],
   s -> 1], {n, 1, nn}]]
Exp[Table[Sum[Log[n/k]*t[n, k], {k, 1, n}], {n, 1, nn}]]