For a few days now I've been trying to find a closed form expression for the determinant of the following $n\times n$ tridiagonal matrix
$$\begin{pmatrix}c_1+b_1+a_1 & b_1 & 0 & \ddots & 0 \\ c_2 & c_2+b_2+a_2 & b_2 & \ddots & 0 \\ 0 & c_3 & c_3+b_3+a_3 & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & b_{n-1}\\ 0 & ... & ... & c_{n} & c_{n}+b_n +a_n\end{pmatrix}$$
For the sequences $c_n$, $b_n$, and $a_n$. I've figured out closed form expression for special cases. Namely, when $a_n=0$, the determinant is $$\Big(\prod_{i=1}^nb_i\Big)\sum_{l=0}^n\prod_{k=1}^l\frac{c_{k}}{b_k}$$ When $l=0$ in the product series, that returns a $1$. Additionally, if $c_1=0$, then the determinant is simply $$\prod_{i=1}^nb_i.$$
I would really like to find an analogous formula in the case where $a_n \neq 0$. For your benefit I will list the first few determinants for small $n$ $$n=1:\quad\quad c_1+b_1+a_1$$ $$n=2:\quad\quad a_1a_2+b_1a_2+a_1b_2+b_1b_2+c_1a_2+c_1b_2+a_1c_2+c_1c_2$$ $$n=3:\quad\quad a_1a_2a_3+b_1a_2a_3+a_1b_2a_3+b_1b_2a_3+a_1a_2b_3+b_1a_2b_3+a_1b_2b_3+b_1b_2b_3+c_1a_2a_3+c_1b_2a_3+c_1a_2b_3+c_1b_2b_3+a_1c_2a_3+a_1c_2b_3+c_1c_2a_3+c_1c_2b_3+a_1a_2c_3+b_1a_2c_3+c_1a_2c_3+a_1c_2c_3+c_1c_2c_3$$
When you look at this, you may suspect that it is just the sum of every $n$th order product of $a$'s $b$'s and $c$'s with no subscript repeated, however this is not the case. For instance, $b_1c_2$ does not appear in the $n=2$ formula. Similarly there are $6$ terms which do not appear in the $n=3$ formula.
I would really appreciate anyones input on this!