Let $(P_n)_n$ be the sequence of polynomials on $\mathbb F_3$ defined by $$P_n=\prod_{\substack{h\in\mathbb F_3[t]\\\deg\,h=n\\h\text{ monic}}}h\qquad (P_0=1).$$ For every $r\in\mathbb N^*$, computations on Maple suggest that $\sum_{i=0}^{r-1}P_i$ is not divisible by any irreducible polynomial of $\mathbb F_3[t]$ with degree $r$.
Is it true in whole generality? If yes any hint or solution to prove that will be welcome. Thanks in advance