When investigating another question regarding matrix let us call it $M_{10}$ I found a peculiar pattern which I can't prove.
We can define $M_n$ to be the $n\times n$ Toeplitz matrix where the elements of the $k$th off diagonal is $k+1$.
It seems to me that at least for 5 or 6 different values of $n$ that $$\det(M_n) = (n+1)\cdot (-1)^{n+1} \cdot 2^{n-2}$$
Maybe recursion can be fruitful? To consider we know $\det(M_n)$ and to try and calculate $\det(M_{n+1})$.
