Closed-form formula for a multivariate polynomial

Question

Counting certain walks in threshold graphs, I came upon the following independent problem. Assume that $x_1,\dots,x_a$ are independent variables and for $k\geq 2$ let $$ P_k(x_1,\dots,x_a)=\sum_{(i_1,\dots,i_k)\in\{1,\dots,a\}^k} x_{i_1}\cdot x_{\min(i_1,i_2)}\cdot x_{\min(i_2,i_3)} \cdots x_{\min(i_{k-1},i_k)} \cdot x_{i_k}. $$ Can you find a closed-form formula for this polynomial, i.e., find the coefficient of the term $x_{j_1}x_{j_2}\cdots x_{j_{k+1}}$ for any given $1\leq j_1\leq j_2\leq\cdots\leq j_{k+1}\leq a$? If this problem appeared somewhere earlier, a reference is more than welcome!