Continuant polynomial

It is defined as follow

K_{n}(x_{1},x_{2},\dots,x_{n}):=\left\{\begin{array}[]{ll}1,&{\rm if~{}}n=0;\\ x_{1},&{\rm if~{}}n=1;\\ x_{1}K_{n-1}(x_{2},\dots,x_{n})+K_{n-2}(x_{3},\dots,x_{n}),&{\rm if~{}}n>1.\\ \end{array}\right.

It is easy to show, that

K_{n}(x_{1},x_{2},\dots,x_{n})=\det\left(\begin{array}[]{rrrrrr}x_{n}&+1&0&% \dots&0&0\\ -1&x_{n-1}&+1&\dots&0&0\\ 0&-1&x_{n-2}&\ddots&0&0\\ \par \vdots&\vdots&\ddots&\ddots&\vdots&\vdots\\ \par 0&0&0&\dots&x_{2}&+1\\ 0&0&0&\dots&-1&x_{1}\\ \end{array}\right).

References: Knuth D.E. ”The Art of Computer Programming, Volume 2: Seminumerical Algorithms.