= Solution
The functions are nonnegative and normalized to have integral one. On the disk $|x|\leq(2n)^{-1/4}$, <Bernoulli inequality> gives $(1-|x|^4)^n\geq1-n|x|^4\geq1/2$. Consequently
$$
c_n^{-1}\geq\frac12\pi(2n)^{-1/2},
\qquad \boxed{c_n\leq\frac{2\sqrt{2n}}{\pi}.}
$$
For any fixed $0<a<1$, the mass outside radius $a$ is at most
$$
\int_{|x|\geq a}g_n(x)\,dx
\leq\pi c_n(1-a^4)^n
\leq2\sqrt{2n}(1-a^4)^n\longrightarrow0.
$$
This is an <approximate identity>. For any test function,
$$
\left|\int g_n\varphi-\varphi(0)\right|
\leq\sup_{|x|\leq a}|\varphi(x)-\varphi(0)|
+2\|\varphi\|_\infty\int_{|x|\geq a}g_n.
$$
Let $n\to\infty$ and then $a\downarrow0$ to conclude $\boxed{g_n\to\delta_0\text{ in }\mathcal D'(\mathbb R^2)}$.
Polar coordinates also give the exact normalization through the <beta function>:
$$
c_n^{-1}=\frac\pi2B(1/2,n+1),\qquad
c_n=\frac{2\Gamma(n+3/2)}{\pi^{3/2}\Gamma(n+1)}.
$$
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2009/iii/paper-9-radial-approximate-identity.png]
{title=Normalized radial densities and cumulative mass concentrating at the origin}
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