Connection between multiplication table $n * k$ and partial sums of the partial sums of the Dirichlet inverse of the Euler totient function.

Question

I am watching this video: L-functions and the Langlands program (RH Saga S1E2)

This reminds me of a recurrence:

Let $c=1$ and a recurrence be:

t[n, k] = 
 If[And[n > 1, k > 1], 
  If[n > k, t[n, k - 1] - t[n - k, k - 1] + t[n - k + c, k], 
   t[k, n - 1] - t[k - n, n - 1] + t[k - n + c, n]], 0]

Then the diagonal sums are tetrahedral numbers:

 {1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364}
 {3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78}
 {3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

https://oeis.org/A000292

Let $c=0$ and apply the same recurrence:

t[n, k] = 
 If[And[n > 1, k > 1], 
  If[n > k, t[n, k - 1] - t[n - k, k - 1] + t[n - k + c, k], 
   t[k, n - 1] - t[k - n, n - 1] + t[k - n + c, n]], 0]

Then the diagonal sums are the partial sums of the partial sums of the Dirichlet inverse of the Euler totient function:

{1, 4, 8, 16, 22, 34, 47, 62, 73, 94, 113, 144}
{3, 4, 8, 6, 12, 13, 15, 11, 21, 19, 31}
{1, 4, -2, 6, 1, 2, -4, 10, -2, 12} *this*

https://oeis.org/A023900

 1, -1, -2, {-1, -4, 2, -6, -1, -2, 4, -10, 2, -12} *this*

---

"c=1 Mathematica program start" 
Clear[nn, t, n, k, c];
nn = 60;
c = 1;
t[n_, 1] = If[n >= 1, n, 0];
t[1, k_] = If[k >= 1, k, 0];
t[n_, k_] := 
  t[n, k] = 
   If[And[n > 1, k > 1], 
    If[n > k, t[n, k - 1] - t[n - k, k - 1] + t[n - k + c, k], 
     t[k, n - 1] - t[k - n, n - 1] + t[k - n + c, n]], 0];
TableForm[Table[Table[t[n, k], {k, 1, nn}], {n, 1, nn}]];
Table[Sum[t[n - k + 1, k], {k, 1, 12}], {n, 1, 12}]
Differences[%]
Differences[%]
"end" 

---

Output:
{3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

---

"c=0 program start" 
Clear[nn, t, n, k, c];
nn = 60;
c = 0;
t[n_, 1] = If[n >= 1, n, 0];
t[1, k_] = If[k >= 1, k, 0];
t[n_, k_] := 
  t[n, k] = 
   If[And[n > 1, k > 1], 
    If[n > k, t[n, k - 1] - t[n - k, k - 1] + t[n - k + c, k], 
     t[k, n - 1] - t[k - n, n - 1] + t[k - n + c, n]], 0];
TableForm[Table[Table[t[n, k], {k, 1, nn}], {n, 1, nn}]];
Table[Sum[t[n - k + 1, k], {k, 1, 12}], {n, 1, 12}]
Differences[%]
Differences[%]
"end" 

---

Output
{-1, -4, 2, -6, -1, -2, 4, -10, 2, -12}

Screen dump 1, multiplication table $n*k$:

Screen dump 2:

https://oeis.org/A030187 Expansion of eta(q) * eta(q^2) * eta(q^7) * eta(q^14) in powers of q.

Question:

Can you prove that the recurrence for the case $c=1$:

t[n, k] = 
 If[And[n > 1, k > 1], 
  If[n > k, t[n, k - 1] - t[n - k, k - 1] + t[n - k + c, k], 
   t[k, n - 1] - t[k - n, n - 1] + t[k - n + c, n]], 0]

gives the multiplication table $n \text{ times } k$?