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We generalize a parametrization obtained by A. V. Corro in (2006) in the three-dimensional Euclidean space. Using this parametrization we study a class of oriented hypersurfaces $M^n$, $n\ge 2$, in Euclidean space satisfying a relation ${\sum }_{r=1}^n{(-1)}^{r+1}r{f}^{r-1}\left(\genfrac{}{}{0pt}{}{n}{r}\right)H_r=0,$ where $H_r$ is the $r$th mean curvature and $f\in C^{\infty }(M^n;ℝ)$, these hypersurfaces are called Weingarten hypersurfaces of the spherical type. This class of hypersurfaces includes the surfaces of the spherical type (Laguerré minimal surfaces). We characterize these hypersurfaces in terms of harmonic...