Clear[nn, t, n, k];
nn = 14;
t[n_, 1] = If[n >= 1, n, 0];
t[1, k_] = 1;
t[n_, k_] := 
  t[n, k] = 
   If[n > k, t[n - k, k], 0] + 
    If[n <= k, 
     Sum[t[n - 1, k - i], {i, 0, n - 1}] - 
      Sum[t[n, k - i], {i, 1, n - 1}], 0];
TableForm[Table[Table[t[n, k], {k, 1, nn}], {n, 1, nn}]];
Flatten[Table[Table[t[n - k + 1, k], {k, 1, n}], {n, 1, nn}]]