Some nice polynomials related to Hankel determinants 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?
 A: 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.
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.
A: This approach uses "Number Walls".
Given a sequence of elements
$\,a:\mathbb{Z}\to F\,$ in a field $\,F.\,$ Define the infinite matrix
of $\,n\times n\,$ Hankel determinants
$$ W_{n,m} :=\det({a_{m-n+i+j}})_{i,j=0}^{n-1}. \tag{1} $$
According to a theorem on determinants by Jacobi, $\,W_{n,m}\,$ satisfies
the Number Wall identity (except at $n=m=0$)
$$ W_{n+1,m}W_{n-1,m} = W_{n,m+1}W_{n,m-1} - W_{n,m}W_{n,m} \tag{2} $$
if $\,a_n=0\,$ for all $\,n<0.\,$
The definition in equation $(1)$ implies
$$ W_{0,m} = 1,\quad W_{1,m} = a_{m-1}. \tag{3} $$
Define the sequence of functions
$$ p_n(k) := W_{k,n+k} = \det({a_{n+i+j}})_{i,j=0}^{k-1}. \tag{4} $$
Use this definition and equation $(3)$ to get
$$ p_n(0) = 1, \quad p_n(1) = a_n. \tag{5} $$
For simplicity, suppose $\,p_n\,$ is a polynomial of degree $\,n\,$ for
$\,n=0,1.\,$ Then equation $(5)$ implies
$$ p_0(x) = a_0 = 1, \quad p_1(x) = x(a_1-1)+1. \tag{6} $$
The definition in equation $(4)$ implies $\, p_0(k) = 1 = W_{k,k}\,$
which is a linear equation in $\,a_{2k}\,$ which can be solved for when
given values of all of the $\,a_0,a_1,\dots,a_{2k-1}.\,$
Similarly, $\,p_1(k) = k(a_1-1)+1 = W_{k,k+1}\,$ which is a linear
equation in $\,a_{2k+1}\,$ which can be solved for when given values of
all the $\,a_0,a_1,\dots,a_{2k}.\,$
Using this procedure, solve the equations successively to get (with $\,z:=a_1$)
$$ a_1 \!=\! z, \;\; a_2 \!=\! 1+z^2\!, \;\; a_3 \!=\!
 2+2z+z^3\!, \;\; a_4 \!=\! 6+4z+3z^2+z^4,\;\;\dots. \tag{7} $$
An alternative method is to rewrite the Number Wall identity equation $(2)$ as
$$ p_{n+1}(k)\,p_{n-1}(k) = p_{n+1}(k-1)\,p_{n-1}(k+1) + p_n(k)\,p_n(k). \tag{8} $$
This implies the $\,p_n(k)\,$ recursion
$$ p_n(k) = (p_n(k-1)\,p_{n-2}(k+1) + p_ {n-1}^2(k))/p_{n-2}(k). \tag{9} $$
Now use equation $(5)$ to get $\,a_n = p_n(1).$
The nice polynomial expression for $\,a_n\,$ is
$$ a_n = \sum_{k=0}^n T_{n,k}\, z^k,\quad
\frac2{1+2x+\sqrt{1-4x}-2xy} = \sum_{n,k} T_{n,k}\, x^n y^k \tag{10} $$
where $\,T\,$ is the triangular array in
OEIS sequence A065600

Triangle T(n,k) giving number of Dyck paths of length 2n with exactly k hills (0 <= k <= n).

For $\,z=0\,$ we get OEIS sequence A000957 with
a different offset.
For $\,z=1\,$ we get $\,a_n=C_n.\,$
For $\,z=2\,$ we get $\,a_n=C_{n+1}.\,$
For $\,z=3\,$ we get $\,a_n=\binom{2n+1}{n}.\,$
For $\,z=4\,$ we get OEIS sequence A049027.
All of these sequences with $\,z>0\,$ are the rows of the infinite array in
OEIS sequence A076037 (with the first column of
all 1s removed).

Note that
$$ p_2(x) = \frac16(1+x)((6-5x+2x^2)-(4x-4x^2)z+(x+2x^2)z^2). \tag{11} $$
This seems to factor into linear factors only if $\,z=1,2,3.$
$$ p_3(x) = \frac1{180}(1+x)(2+x)(3+2x)F_3(x,z) \tag{12} $$ where
$\,F_3(x,z)\,$ is a cubic in $\,x\,$ and $\,z\,$ which also seems to
factor into linear factors only if $\,z=1,2,3.$
$$ p_4(x) = \frac1{75600}(1+x)(2+x)^2(3+x)(3+2x)(5+2x)F_4(x,z) \tag{13} $$ where
$\,F_4(x,z)\,$ is degree $4$ in $\,x\,$ and $\,z.\,$
It seems to factor into linear factors only if $\,z=1,2,3.$
This is consistent with the definitions of $\,f_n(x)\,$ and $\,g_n(x)\,$
given in the question.
