(A comment rather than an answer.)
Here is a plot of $a_n/n^5$ (red) and $b_n/n^5$ (blue). It might not go far enough to show the asymptotic behaviour, but a possibility is that $a_n$ and $b_n$ are asymptotically the same and of order $\Theta(n^5)$.
Here is a plot of $a_n/b_n$ with odd $n$ in red and even $n$ in blue.
