Prove that the entries in this matrix essentially consists of the Dirichlet inverse of the Euler totient.

Question

The identity matrix satisfies the following recurrence:

(*Mathematica 8*)
Clear[t];
nn = 15;
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], 
     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]];
a = Table[Table[t[n, k], {k, 1, nn}], {n, 1, nn}];
MatrixForm[a]

Prove that the matrix: $$T(n,k)=a(GCD(n,k))$$ where GCD is the Greatest Common Divisor and the sequence a is the Dirichlet Inverse of the Euler totient function, for $n>1$ and $k>1$, satisfies the following recurrence:

Clear[t];
nn = 15;
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], 
     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]];
a = Table[Table[t[n, k], {k, 1, nn}], {n, 1, nn}];
MatrixForm[a]

The only difference between the two programs is that the upper summation limits are different.