  We study arithmetic and asymptotic properties of polynomials provided by
$Q_n(x):= x \sum_{k=1}^n k \, Q_{n-k}(x)$ with initial value $Q_0(x)=1$. The
coefficients satisfy a central limit theorem and a local limit theorem
involving Fibonacci numbers. We apply methods of Berry and Esseen, Harper,
Bender, and Canfield.
