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.