I am looking for the determinant

$$ \det(I_n + H_n) $$

where $I_n$ is the $n \times n$ identity matrix and $H_n$ is the $n \times n$ Hilbert matrix, whose entries are given by

$$ [H_n]_{ij} = \frac{1}{i+j-1}, \qquad\qquad 1 \le i,j \le n $$

Is anything known about this determinant for finite $n$ or about its asymptotic behaviour for $n \rightarrow \infty$?

More generally, are there results about the determinant of "identity plus Hankel" matrices or their asymptotic behaviour?