This is the Hankel determinant associated to the sequence $m_n = \mathbb{E}(X^n) = n!$ of moments of an exponential distribution with mean $1$. Some general results can be used to show that the sequence of Hankel determinants associated to the moments of a random variable are always positive iff the induced measure on $\mathbb{R}$ has infinite support, and some more general results can be used to exactly calculate the Hankel determinants as in the comments by calculating an appropriate sequence of orthogonal polynomials. The relevant orthogonal polynomials for the exponential distribution are the Laguerre polynomials.

I gather this is very classical material but I don't know a reference; you can see a writeup in slightly unusual language here.

This is the Hankel determinant associated to the sequence $m_n = \mathbb{E}(X^n) = n!$ of moments of an exponential distribution with mean $1$. Some general results can be used to show that the sequence of Hankel determinants associated to the moments of a random variable are always positive iff the induced measure on $\mathbb{R}$ has infinite support, and some more general results can be used to exactly calculate the Hankel determinants as in the comments by calculating an appropriate sequence of orthogonal polynomials. The relevant orthogonal polynomials for the exponential distribution are the Laguerre polynomials.

I gather this is very classical material but I don't know a reference; you can see a writeup in slightly unusual language here.

Edit: Ah, here's a reference: Section 2.7 of Krattenthaler's Advanced Determinant Calculus.