Log convexity (meaning $a_k^2\leq a_{k-1}a_{k+1}, \ 1\leq k\leq n-1$) seems to imply your upside down unimodal condition, and according to this [paper][1], there are several such combinatorial sequences.  An easy example of a log convex sequence is  the factorial sequence
$$(0!, 1!, \ldots, n!);$$
it's upside down unimodal with $m=0$ (but then again, I guess every monotone sequence trivially satisfies both right side up and upside down unimodality conditions).  In fact, it seems that your sequence above $(b_{n,0},\ldots,b_{n,n})$ is log convex as well. 

  [1]: https://www.sciencedirect.com/science/article/pii/S0196885806002016