Say $X_{ij}$ are i.i.d. random variables with known moments $\mathbb{E}\left[X_{ij}^n\right] = \mu_n$. Given a random matrix $A = \{X_{ij}\} _{n\times n}$, what is the expected value
$$V_m = \mathbb{E}\left[(\det A)^m\right]$$
equal to? (Where $m$ being a positive integer) By the antisymmetry of a determinant, it is trivial that $V_m$ is $0$ for all odd values of $m$. I was courious whether a general formula can be found for $m\geq 4$, since by an ingenious argument (see this or this mathoverflow article), we are able to find the exact result for $m=2$:
$$V_2 = n! (\mu_2-\mu_1^2)^{n-1}(\mu_2+\mu_1^2(n-1))$$
A special interest is for $X_{ij}$ being exponentialy distributed, i.e. with $\mu_n = n!$