Nontrivial Solutions of a Higher-Order Rational Difference Equation

We prove that, for every $k\in\mathbb N$, the following generalization of the Putnam difference equation
$$ x_{n+1}=\frac{x_n+x_{n-1}+\dots+x_{n-(k-1)}+x_{n-k}x_{n-(k+1)}} {x_nx_{n-1}+x_{n-2}+\dots+x_{n-(k+1)}}\,,\qquad n\in\mathbb N_0, $$
has a positive solution with the following asymptotics
$$ x_n=1+(k+1)e^{-\lambda^n}+(k+1)e^{-c\lambda^n}+o(e^{-c\lambda^n}) $$
for some $c>1$ depending on $k$, and where $\lambda$ is the root of the polynomial $P(\lambda)=\lambda^{k+2}-\lambda-1$ belonging to the interval $(1,2)$. Using this result, we prove that the equation has a positive solution which is not eventually equal to $1$. Also, for the case $k=1$, we find all positive eventually equal to unity solutions to the equation.