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"](https://arxiv.org/pdf/math/0104241.pdf) was motivated by [Somos sequences](https://en.wikipedia.org/wiki/Somos_sequence). 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\*}



"Laurent phenomenon"?
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.


