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Brendan McKay
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(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

Brendan McKay
  • 37.7k
  • 3
  • 80
  • 147