Let $(a_n)$ be the sequence defined by $$a_{n+1}=2na_n-n^2a_{n-1}$$ and $a_0=0$ and $a_1=1$. I would like to prove that there exist infinitely many primes $p$ such that $p$ does not divide $a_p$. Any help or reference will be welcome.
Thanks in advance
edit: I made a mistake for the first terms: $a_0=0$ and $a_1=1$.