The following symmetric matrix $A$ is given. Let denote $A_i$ the $i$-th leading principal submatrix of the matrix $A$.
$$ A = \begin{bmatrix} a_1 & b_1 \\ b_1 & a_2 & b_2 \\ & b_2 & \ddots & \ddots \\ & & \ddots & \ddots & b_{n-1} \\ & & & b_{n-1} & a_n \end{bmatrix} $$
The determinant of the matrix $A$ is given by the recurrence relation
$det(A_i) = a_i det(A_{i-1}) - b_{i-1}^2det(A_{i-2})$
I would like to know is there any other relation among the determinant of the leading principal submatrix of $A$? for example a numerically that determines determinant of $A_j$ is greater than $A_m$ for $j > m$ or any thing else.