Number theoretic analogue of the identity matrix by permuting upper summation indexes.

(*start*)
(*Mathematica*)
Clear[t];
nn = 15;
mm = 15;
t[1, 1] = 1;
t[n_, k_] :=
  t[n, k] =
   If[n < k,
    If[And[n > 1, k > 1],
     Sum[t[k - i, n - 1] - t[k - i, n], {i, 1, k - 1}], 0],
    If[And[n > 1, k > 1],
     Sum[t[n - i, k - 1] - t[n - i, k], {i, 1, n - 1}], 0]];
TableForm[Table[Table[t[n, k], {k, 1, mm}], {n, 1, nn}]]

Clear[t];
nn = 15;
mm = 15;
t[1, 1] = 1;
t[n_, k_] :=
  t[n, k] =
   If[n < k,
    If[And[n > 1, k > 1],
     Sum[t[k - i, n - 1] - t[k - i, n], {i, 1, n - 1}], 0],
    If[And[n > 1, k > 1],
     Sum[t[n - i, k - 1] - t[n - i, k], {i, 1, k - 1}], 0]];
TableForm[Table[Table[t[n, k], {k, 1, mm}], {n, 1, nn}]]
(*end*)