Math experiment done with Mathematica

```
f[n_] := NMaximize[{Evaluate[RealAbs[D[Exp[-1/x^2], {x, n}]]]/
n!^(3/2), RealAbs[x] <= 1 && n >= 1 && n \[Element] PositiveIntegers}, {x},
Method -> {"DifferentialEvolution", "ScalingFactor" -> 1},
AccuracyGoal -> 5, PrecisionGoal -> 5, WorkingPrecision -> 15]
Table[f[k], {k, 1, 20}]
```

results in
$$\left(
\begin{array}{cc}
0.819832557883720 & \{x\to -0.816496580926071\} \\
1.05309210878418 & \{x\to -0.520851654621382\} \\
1.43213203276870 & \{x\to -0.403266928496221\} \\
2.02929498643239 & \{x\to -0.487430660529263\} \\
3.99625841701811 & \{x\to -0.397684505142533\} \\
7.32790921886967 & \{x\to -0.341047324550614\} \\
12.8456602395717 & \{x\to -0.301641954804728\} \\
20.1279218513637 & \{x\to 0.333941189526955\} \\
40.8839394652186 & \{x\to 0.299174580083694\} \\
79.3007302990450 & \{x\to -0.272480728040435\} \\
148.468049756260 & \{x\to 0.251254191655359\} \\
270.267689843771 & \{x\to 0.233912856932535\} \\
491.924121098429 & \{x\to -0.249799234272746\} \\
976.765593945151 & \{x\to -0.233559061606742\} \\
1885.77886324325 & \{x\to -0.219827966408129\} \\
3555.54552839170 & \{x\to 0.208043856897294\} \\
6569.30249774506 & \{x\to 0.197802964493932\} \\
11926.3007981150 & \{x\to 0.188807540307310\} \\
11115.5329809164 & \{x\to 0.165430061926195\} \\
48517.0571216129 & \{x\to -0.189175462374212\} \\
\end{array}
\right) ,$$ suggesting $a=1$ and $n_0=2$.

Addition. A numerical proof with Mathematica can be done as follows.

```
f[n_?NumericQ] := NMaximize[{RealAbs[Evaluate[D[Exp[-1/x^2], {x, n}]]]/n!^(3/2),
RealAbs[x] <= 1 && n >= 1 &&
n \[Element] PositiveIntegers}, {x},][[1]]
NMinimize[{Evaluate[f[n]], n >= 2 && n \[Element] Integers}, n,
Method -> {"DifferentialEvolution", "ScalingFactor" -> 1},
AccuracyGoal -> 5, PrecisionGoal -> 5, WorkingPrecision -> 15]
```

$$\{1.05309210878418,\{n\to 2\}\} $$

It should be noticed that the execution of the above code takes a lot of time.