(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 $\Theta(n^5)$.

[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/l6FOa.png