zeta squared relation to von Mangoldt function matrix

(*start*)
Clear[t, a, b];
a = 0;
b = 1;
nn = 12;
t[1, 1] = 1;
t[n_, 1] = 0;
t[1, k_] = 0;
t[n_, k_] :=
  t[n, k] =
   If[n < k,
    If[And[n > 1, k > 1],
     a*Sum[t[k - i, n - 1] - t[k - i, n], {i, 1, n - 1}], 0],
    If[And[n > 1, k > 1],
     b*Sum[t[n - i, k - 1] - t[n - i, k], {i, 1, k - 1}], 0]];
a = Table[Table[t[n, k], {k, 1, nn}], {n, 1, nn}];
MatrixForm[a]

Clear[t, a, b];
a = 1;
b = 1;
nn = 12;
t[1, 1] = 1;
t[n_, 1] = 0;
t[1, k_] = 0;
t[n_, k_] :=
  t[n, k] =
   If[n < k,
    If[And[n > 1, k > 1],
     a*Sum[t[k - i, n - 1] - t[k - i, n], {i, 1, n - 1}], 0],
    If[And[n > 1, k > 1],
     b*Sum[t[n - i, k - 1] - t[n - i, k], {i, 1, k - 1}], 0]];
a = Table[Table[t[n, k], {k, 1, nn}], {n, 1, nn}];
MatrixForm[a]
(*end*)