How to find the $m^\text{th}$ term for the following expression:
$$ \left.\frac{\partial^m}{\partial s^m}e^{a s^2}\right|_{s=0}$$
Is there any analytical approach?
I computed first few terms and used mathematica "FindSequenceFunction", which yielded the $m^\text{th}$ term as (edited expression):
$$ \frac{2^{m-1} \left(1+(-1)^{m}\right) a^{\frac{m}{2}} \Gamma \left(\frac{m+1}{2}\right)}{\sqrt{\pi }}$$
The first few terms:
$1,0,2 a,0,12 a^2,0,120 a^3,0,1680 a^4,0,30240 a^5,0,665280 a^6,0,17297280 a^7,0,518918400 a^8,0,17643225600 a^9,0,670442572800 a^{10}$
Any help will be appreciated.
Since$$e^{as^2}=1+as^2+\frac{a^2}{2!}s^4+\frac{a^3}{3!}s^6+\cdots$$you have$$\left.\frac{\mathrm d^m}{\mathrm d^ms}e^{as^2}\right|_{s=0}=\begin{cases}m!\dfrac{a^{\frac m2}}{\left(\frac m2\right)!}&\text{ if $m$ is even}\\0&\text{ if $m$ is odd.}\end{cases}$$