The determinant of a matrix

Question

In order get the determinant of$$\begin{pmatrix} \lambda-n-1 & 1 & 2 & 2 & 1 & 1 & 1& 1 & \cdots &1 & 1 \\ 1 & \lambda-2n+4 & 1 & 2 & 2 &2 &2&2 & \cdots &2&2 \\ 2&1&\lambda-3n+9&1&2&3&3&3&\cdots & 3 &3\\ 2&2&1&\lambda-3n+9&1&3&3&3&\cdots&3&3\\ 1&2&2&1&\lambda-2n+4&2&2&2&\cdots&2&2\\ 1&2&3&3&2&\lambda-2n+1&2&2&\cdots&2&2\\ 1&2&3&3&2&2&\lambda-2n+1&2&\cdots&2&2\\ 1&2&3&3&2&2&2&\lambda-2n+1&\cdots&2&2\\ \cdots&&\cdots&&\cdots&&\cdots&&\cdots&&\cdots\\ 1&2&3&3&2&2&2&2&\cdots&\lambda-2n+1&2\\ 1&2&3&3&2&2&2&2&\cdots&2&\lambda-2n+1 \end{pmatrix},$$ it seems that a lot of calculation is required. How to get it in a comparably faster way? Thanks.