(*Mathematica 8*)
Clear[t]
K = 2
nn = 12;
t[n_, 1] = If[n >= 1, n, n];
t[n_, k_] :=
 t[n, k] =
  If[n >= k,
   Sum[t[(n - i), k - 1], {i, 1, K*(k - 1)}] -
    Sum[t[(n - i), k], {i, 1, K*(k - 1) + 1}], n/k]
TableForm[Table[Table[t[n, k], {k, 1, nn}], {n, 1, nn}]]
TableForm[Table[Table[t[n, k] - n/k, {k, 1, nn}], {n, 1, nn}]]