Let $M_n$ be the matrix $$M_n=\begin{pmatrix} 1&\binom{1}{1}\binom{1-1}{1-1} &0 &0\qquad \qquad \dots &0\\ 1&\binom{2}{1}\binom{2-1}{1-1} &\binom{2}{2}\binom{2-1}{2-1} &0 \qquad\qquad\dots & 0\\ 1&\binom{3}{1}\binom{3-1}{1-1} &\binom{3}{2}\binom{3-1}{2-1} &\binom{3}{3}\binom{3-1}{3-1}\qquad\dots &0\\ \qquad \qquad \dots\\ \qquad \qquad \dots\\ 1&\binom{n}{1}\binom{n-1}{1-1} &\binom{n}{2}\binom{n-1}{2-1} &\binom{n}{3}\binom{n-1}{3-1}\qquad\dots &\binom{n}{n-1}\binom{n-1}{n-2}\end{pmatrix}.$$
Question. It appears that the modified Bessel function of the first kind, of order $0$, has the expansion $$I_0(x)=\exp\left(\sum_{j=1}^{\infty}\frac{(-1)^{j-1}\det(M_j)}{j!^2}\left(\frac{x}2\right)^{2j}\right).$$ Is this true?
