  We study the zero distribution of non-orthogonal polynomials attached to
$g(n)=s(n)=n^2$: \begin{equation*} Q_n^g(x)= x \sum_{k=1}^n g(k) \,
Q_{n-k}^g(x), \quad Q_0^g(x):=1. \end{equation*} It is known that the case
$g=id$ involves Chebyshev polynomials of the second kind. The zeros of
$Q_n^s(x)$ are real, simple, and are located in $(-6\sqrt{3},0]$. Let
$N_n(a,b)$ be the number of zeros between $-6 \sqrt{3} \leq a < b \leq 0$. Then
we determine a density function $v(x)$, such that \begin{equation*} \lim_{n
\rightarrow \infty} \frac{N_n(a,b)}{n} = \int_a^b v(x) \,\, \mathrm{d}x.
\end{equation*} The polynomials $Q_n^s(x)$ satisfy a four-term recursion. We
present in detail an analysis of the fundamental roots and give an answer to an
open question on recent work by Adams and Tran--Zumba. We extend a method
proposed by Freud for orthogonal polynomials to more general systems of
polynomials. We determine the underlying moments and density function for the
zero distribution.
