The determinant of the matrix whose entries are $\gcd(i,j)$ for $1≤i,j≤n$ equals $\prod_{k=1}^n \varphi(k)$ where $\varphi$ is Euler's totient function: see A001088 in the OEIS, as well as the paper “The determinants of GCD matrices” by Zhongshan Li, Linear Algebra Appl. 134 (1990) 137–143 and the references it contains regarding the determinant of the matrix whose entries are $\gcd(x_i,x_j)$ for $\{x_1,\ldots,x_n\}$ a factor-closed set of positive integers (we get $\prod_{k=1}^n \varphi(x_k)$ up to sign).
Question: What about the determinant of the matrix whose entries are $\gcd(i,j)$ for $0≤i,j≤n-1$ instead? (I.e., labeling rows and columns from $0$ instead of $1$. Note that the result I just cited doesn't apply. Also, just to be clear, $\gcd(0,j) = j$ for all $j$.) Changing the sign, the first values of this determinant are $$ 0, 1, 2, 8, 24, 160, 384, 3456, 16896, 129024\ldots $$ This sequence is not in the OEIS. Does it have a simple expression involving the $\varphi(k)$ (it appears to have some resemblance to the $\prod_{k=1}^n \varphi(2k)$, but it's not that)?