Let
$f\_n(x)=\prod\_{j=0}^{\lfloor{\frac{n-1}{2}}\rfloor}\prod\_{i=2j+1}^{2n-2j-1}\frac{2x+i}{i}$
and
$g\_n(x)=\prod\_{j=1}^{\lfloor{\frac{n}{2}}\rfloor}\prod\_{i=2j}^{2n-2j}\frac{2x+i}{i}.$


Then
$f\_n(k)=\det \left( {f\_{n+i+j}(1) } \right)\_{i,j = 0}^{k - 1}$ for each positive integer $k$ and analogously for $g\_n(k).$


Note that $f\_n(1)=\binom{2n+1}{n}$ and $g\_n(1)=C\_n,$ a Catalan number.


Do these polynomials also occur in other contexts? Are there other integer sequences $a\_n$


such that the polynomials $p\_n(x)$ with $p\_n(k)=\det \left( {a\_{n+i+j} } \right)\_{i,j = 0}^{k - 1}$ are nice?



Some nice polynomials related to Hankel determinants
See Example 4 in Section 3.1.6 of <https://arxiv.org/abs/1409.2562> (Federico Ardila, "Algebraic and geometric methods in enumerative combinatorics").


It explains that your $g\_n(k)$ has an interpretation in terms of nested fans of Dyck paths.


Equivalently, as I mentioned in a comment above, $g\_n(k)$ is the order polynomial of the (unshifted) staircase partition $(n,n-1,\ldots,1)$ shape poset. In other words, it is the number of ways to fill the shape
$$ \begin{array}c \square & \square & \square & \square \\ \square & \square & \square \\ \square & \square \\ \square \end{array} $$
with nonnegative integers between $0$ and $k$ that are weakly decreassing in rows and columns.


I explained something similar in a [previous MO question of yours](https://mathoverflow.net/questions/372811/number-of-bounded-dyck-paths-with-negative-length-as-hankel-determinants).


EDIT: Since this question got bumped, let me edit it to mention also that there is also an analogous interpretation of your $f\_n(k)$, in terms of the shifted double staircase shape poset, as explained in more detail in my comment: [Generating functions for Hankel determinants of Catalan numbers](https://mathoverflow.net/questions/402767/generating-functions-for-hankel-determinants-of-catalan-numbers#comment1030233_402795).


