# 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?