Is there a formula to compute the determinant of a structurally symmetric $n$-banded matrix? I am specifically interested in the 5-banded matrix:

$$ \left[\begin{matrix} c_{0} & s_{0} & 0 & w_{0} & 0 & 0 & 0 & 0 & 0\\ n_{0} & c_{1} & s_{1} & 0 & w_{1} & 0 & 0 & 0 & 0\\ 0 & n_{1} & c_{2} & 0 & 0 & w_{2} & 0 & 0 & 0\\ e_{0} & 0 & 0 & c_{3} & s_{3} & 0 & w_{3} & 0 & 0\\ 0 & e_{1} & 0 & n_{3} & c_{4} & s_{4} & 0 & w_{4} & 0 & \dots \\ 0 & 0 & e_{2} & 0 & n_{4} & c_{5} & 0 & 0 & w_{5}\\ 0 & 0 & 0 & e_{3} & 0 & 0 & c_{6} & s_{6} & 0\\ 0 & 0 & 0 & 0 & e_{4} & 0 & n_{6} & c_{7} & s_{7}\\ 0 & 0 & 0 & 0 & 0 & e_{5} & 0 & n_{7} & c_{8} \\ & & & & \vdots & & & & & \ddots\end{matrix}\right] $$

which is defined by the five sequences:

$$ \left\lbrace w_0, w_1, \cdots \right\rbrace, \left\lbrace s_0, s_1, \cdots \right\rbrace, \left\lbrace c_0, c_1, \cdots \right\rbrace, \left\lbrace n_0, n_1, \cdots \right\rbrace, \left\lbrace e_0, e_1, \cdots \right\rbrace $$

All other elements are zero.

It's not clear to me whether formulas for tridiagonal matrices can be extended straightforwardly to compute the determinant of the above matrix.

Notes:

- In my special case some terms $ n_2 = n_5 = n_8 = \cdots = 0 $, $ s_0 = s_3 = s_6 = \cdots = 0 $, but I'm interested in the solution for arbitrary sequences $w_i$, $s_i$, $c_i$, $n_i$, $e_i$.
- All off-diagonal elements are nonnegative, and all diagonal elements are negative.