  We study nonnegative solutions to the following Hardy-H\'enon type equations
involving higher order fractional Laplacians $$ (-\Delta)^\sigma u =
|x|^{-\alpha}u^{p} ~~~~~~ \mbox{in} ~ \mathbb{R}^n \backslash \{0\} $$ with a
possible 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 direct
approach without using the super poly-harmonic properties, we establish an
integral representation for nonnegative solutions to the above higher order
fractional equations whether the singularity $\{0\}$ is removable or not. As
the first application, we prove an optimal Liouville-type theorem for the above
equations with removable singularity for all $\sigma \in (0, n/2)$ when $$ 1 <
p < p_{\sigma,\alpha}^*:=\frac{n+2\sigma -2\alpha}{n-2\sigma} ~~~ \mbox{and}
~~~ -\infty < \alpha < 2\sigma. $$ This, in particular, 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
solutions in the critical case or in the case when the origin is a
non-removable singularity. Such radial symmetry would be useful in studying the
singular Yamabe-type problems.
