What could be these polynomials? Trying to build an approximation, I encountered the following polynomials
$$\left(
\begin{array}{cc}
 1 & 1 \\
 2 & x-3 \\
 3 & x^2-30 x+45 \\
 4 & x^3-273 x^2+1575 x-1575 \\
 5 & x^4-2460 x^3+43470 x^2-132300 x+99225 \\
 6 & x^5-22143 x^4+1123650 x^3-8004150 x^2+16372125 x-9823275 \\
 7 & x^6-199290 x^5+28423395 x^4-431531100 x^3+1830403575 x^2-2809456650 x+1404728325
\end{array}
\right)$$
What I noticed is that

*

*the constant term is $(-1)^{n}\,\frac{2^{2 n+1} \Gamma \left(n+\frac{1}{2}\right) \Gamma
   \left(n+\frac{3}{2}\right)}{\pi }$

*the coefficient of the second highest power is $\frac{3}{8} \left(1-9^n\right)$

*the coefficient of the third highest power is $\frac{15}{128} \left(2-3^{2 n+5}+5^{2 n+4}\right)$ but this does not help me much.

Any idea or suggestion would be very welcomed

$\{-1575, -132300, -8004150, -431531100,...\}$:
$\dfrac{35}{1024}\bigg(5 - 3^{2 n + 8} + 5^{2 n + 7} - 7^{2 n + 6}\bigg)$
$\{1, -30, 1575, -132300, 16372125, -2809456650,...\}$:
$-(-1)^{n + 1} 2^{2 n} (n + 1) Pochhammer\bigg(\dfrac{3}{2}, n\bigg) Pochhammer\bigg(\dfrac{5}{2}, n\bigg)$
