reference request – Some nice polynomials related to Hankel determinants

Let
$f_n(x)=prod_{j=0}^{n}prod_{i=2j+1}^{2n-2j-1}frac{2x+i}{i}$
and
$g_n(x)=prod_{j=1}^{n-1}prod_{i=2j}^{2n-2j}frac{2x+i}{i}.$

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?