Consider the following generating function :
$$\sum_{k=0}^\infty\sum_{n=0}^\infty\sum_{m=0}^\infty \frac {n^{2m+4}(-1)^m\Gamma(2k+1,-(am+b))}{m!(am+b)^{2k+1}} x^{2k}$$
Where , $\Gamma(p,q)$ is incomplete gamma function
$a,b,$ are constants .
Question :How to get a closed form of this generating function ?