Binomial expansion of tau number of divisors

(*Mathematica*)Clear[t, n, k, x, c, y, z, w, p, q, r, s, x1];
nn = 8;
t[n_, 1] := t[n, 1] = 1;
t[n_, k_] :=
  t[n, k] =
   If[n >= k,
    Sum[t[n - i, k - 1]*ToExpression[StringJoin["y", ToString[i]]] +
      t[n - i, k]*ToExpression[StringJoin["x", ToString[i]]], {i, 1,
      k - 1}], 0];
TableForm[Table[Table[t[n, k], {k, 1, nn}], {n, 1, nn}]]
TableForm[Table[Table[Expand[t[n, k]], {k, 1, nn}], {n, 1, nn}]]