The Padovan polynomials $P_n(x)$ are defined by $$P_n(x) = \left\{ \begin{array}{ll} 1 &; n=1 \\ 0 &; n=2 \\ x &; n=3 \\ xP_{n-2}(x)+P_{n-3}(x)&; n \geq 4. \end{array} \right.$$