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We study nonnegative solutions to the following Hardy-H\'enon type equationsinvolving higher order fractional Laplacians $$ (-\Delta)^\sigma u =|x|^{-\alpha}u^{p}  in ~~ \mathbb{R}^n \backslash \{0\} $$ which have apossible singularity at the origin, where $\sigma$ is a real number satisfying$0 < \sigma < n/2$, $-\infty < \alpha < 2\sigma$ and $p>1$. By a more directapproach without using the super poly-harmonic properties of nonnegativesolutions, we establish an integral representation for nonnegative solutions tothe above higher order fractional equations whether the singularity $\{0\}$ isremovable or not. As the first application, we prove Liouville theorems fornonnegative solutions to the above equations with removable singularity for all$\sigma \in (0, n/2)$ when $$ 1 < p < p_{\sigma,\alpha}^*:=\frac{n+2\sigma-2\alpha}{n-2\sigma}~~~ and ~~~ -\infty < \alpha < 2\sigma. $$ This, inparticular, covers a gap occurring for non-integral $\sigma \in (1, n/2)$ and$\alpha \in (0, 2\sigma)$ in the current literature. As the second application,we show the radial symmetry of nonnegative solutions to the above equations inthe critical case or in the case where the singularity at the origin is notremovable. Such radial symmetry would be useful in studying the higher ordersingular Yamabe problems and characterizing isolated singularities of thehigher order Hardy-H\'enon equations.

Liouville-type theorems, radial symmetry and integral representation of  solutions to Hardy-H\'enon equations involving higher order fractional  Laplacians