<s>[INST] "Laurent phenomenon"? Define the recurrence
\begin{align*}
n(2n+x-3)u_n(x)
&=2(2n+x-2)(4n^2+4nx-8n-3x+3)u_{n-1}(x) \\
&-4(n+x-2)(2n-3)(2n+2x-3)(2n+x-1)u_{n-2}(x)
\end{align*}
with initial conditions $u_0(x)=0$ and $u_1(x)=x+1$.
The subject of "Laurent phenomenon" was motivated by Somos sequences. In the same spirit, I ask:

QUESTION. Is it true that each $u_n(x)$ is a polynomial in $x$? In fact, with positive integer coefficients.

EXAMPLES. $u_2(x)=5x^2 + 13x + 6$ and $u_3(x)=22x^3 + 114x^2 + 164x + 60$ and
\begin{align*}
u_4(x)&=93x^4 + 814x^3 + 2367x^2 + 2606x + 840 \\
u_5(x)&=386x^5 + 5140x^4 + 25030x^3 + 54500x^2 + 51024x + 15120.
\end{align*}
 [/INST] In fact,
$$
u_n(x) = 
{2}^{n-1}\prod _{k=0}^{n-1}(2x+2k+1)
-{2\,n-1\choose n-1}\prod _{k=0}^{n-1}(x+k) ,
\tag1$$
which is a polynomial with integer coefficients.
P.S. the proof rests on a routine verification that (1) satisfies the given recurrence relation and initial conditions.
</s>