(*Mathematica start*)
"Sum recurrence:"
Clear[t, T, n, k, i];
nn = 8;
t[n_, 1] = If[n >= 1, 0, 0];
t[n_, k_] := 
 t[n, k] = 
  If[n >= k, (Sum[t[n - i, k - 1], {i, 1, k - 1}] - 
     Sum[t[n - i, k], {i, 1, k - 1}]), 1]
TableForm[T = Table[Table[t[n, k], {k, 1, nn}], {n, 1, nn}]]
"Product recurrence:"
Clear[t, T, n, k, i];
nn = 8;
t[n_, 1] = If[n >= 1, 0, 0];
t[n_, k_] := 
 t[n, k] = 
  If[n >= k, (Product[t[n - i, k - 1], {i, 1, k - 1}] - 
     Product[t[n - i, k], {i, 1, k - 1}]), 1]
TableForm[T = Table[Table[t[n, k], {k, 1, nn}], {n, 1, nn}]]
(*Mathematica end*)