= Solution
Put $w=1/z$ and $Y(w)=y(1/w)$. The equation becomes
$$
Y''+\frac2wY'-w^{-n-4}Y=0.
$$
At $w=0$ the regular-singular criterion requires $w(2/w)$ and $w^2(-w^{-n-4})$ to be analytic. This holds precisely for $n\le-2$. Thus the singularity at infinity is irregular, corresponding to essential exponential behaviour of solutions, \b[exactly for $\boxed{n\ge-1}$]. For $n\le-2$ it is regular singular, with powers and possibly logarithms rather than such essential exponentials.
For $y''=z^3y$, the <Liouville–Green approximation> uses $Q=z^3$, amplitude $Q^{-1/4}$ and phase integral $\int\sqrt Q\,dz=2z^{5/2}/5$. On a consistently chosen branch, the two leading solutions are
$$
\boxed{y_\pm(z)\sim z^{-3/4}\exp\!\left(\pm\frac25z^{5/2}\right).}
$$
The derivative correction is smaller by order $z^{-5/2}$, and the two exponential signs give independent solutions.
Using the convention that <Stokes lines> have equal exponential phase, $\operatorname{Im}(z^{5/2})=0$, their rays are $\arg z=2\pi k/5$. The equal-magnitude, oscillatory rays, often called <anti-Stokes lines>, satisfy $\operatorname{Re}(z^{5/2})=0$, giving $\arg z=(2k+1)\pi/5$. If the alternative naming convention is used, these two names interchange; the stated equations specify the rays unambiguously.
To give the sector precisely, the solution recessive on the positive axis can be represented by
$$
y_0(z)=\frac2{\sqrt{5\pi}}\sqrt z\,K_{1/5}\!\left(\frac25z^{5/2}\right),
$$
where $K_\nu$ is the <Modified Bessel function of the second kind>. Substitution verifies the differential equation. Its asymptotic $K_\nu(w)\sim\sqrt{\pi/(2w)}e^{-w}$ is valid on closed subsectors of $|\arg w|<3\pi/2$ on the continued branch. Consequently
$$
\boxed{y_0(z)\sim z^{-3/4}e^{-2z^{5/2}/5}\quad\text{for }|\arg z|<3\pi/5.}
$$
Actual exponential decay occurs in the narrower sector $|\arg z|<\pi/5$; in the adjacent portions of the larger asymptotic sector the same continued exponential grows. This distinguishes the validity of the approximation to the positive-axis recessive solution from the directions where its modulus decays.
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