jmc

We have $$\begin{array}{rl}{P}_1(x)& =P_0(x-1),\\ P_2(x)& =P_1(x-2)=P_0(x-2-1),\\ P_3(x)& =P_2(x-3)=P_0(x-3-2-1),\end{array}$$and so on. We see that $$\begin{array}{rl}{P}_{20}(x)& =P_0(x-20-19-\cdots -2-1)\\ & =P_0(x-210),\end{array}$$using the formula $20+19+\cdots +2+1=\frac{20(21)}{2}=210.$ Thus, $$P_{20}(x)=(x-210{)}^3+313(x-210{)}^2-77(x-210)-8.$$The coefficient of $x$ in this polynomial is $$\begin{array}{rl}3\cdot 210^2-313\cdot 2\cdot 210-77& =210(3\cdot 210-313\cdot 2)-77\\ & =210(630-626)-77\\ & =210\cdot 4-77\\ & =\boxed{763}.\end{array}$$