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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, 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.