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.
Added October 10, 2023: I stumbled across the following result (Theorem 9) in this old paper of Stanley on log concavity and unimodality, attributed to Rees and Sharp:
Theorem: (Rees-Sharp 1978) Let $R$ be a commutative Noetherian local ring with maximal ideal $\mathfrak{m}$ and Krull dimension $n$, and let $I,J\subset R$ be any $\mathfrak{m}$-primary ideals, and let $\ell(R/I^rJ^s)$ be length of the quotient of $R$ by powers of the product of ideals. Then for $r$ and $s$ sufficiently large, $\ell(R/I^rJ^s)$ is a polynomial in $r$ and $s$, and if the total degree $n$ part of that polynomial is $$P(r,s)=\frac{1}{n!}\sum_{k=0}^n\binom{n}{k}E_k(I,J)r^{n-k}s^k$$ then the coefficient sequence $E_0(I,J),\ldots,E_n(I,J)$ is log convex, and each $E_k(I,J)$ is a nonnegative integer (I think these are called mixed multiplicities).