Does the following expression has a closed form \begin{align} E \left[ \| Z\|^k \exp(t \|Z\|^2)\right] \end{align} for $k$ is even and $Z$ is standard normal vector.
For the case of $k=2$ the computation is very easy. For $k=1/2$ see here Compute $E[ \sqrt{Q} e^{-tQ} ]$ where $Q$ is non-central chi-square random variable..
However, this question is about even $k$. I there a way to use a binomina theorem to solve this ?
In hyperspherical coordinates in $\mathbb R^n$, the integral becomes $$\operatorname E \left[ \lVert Z \rVert^k e^{t \lVert Z \rVert^2} \right] = (2 \pi)^{-n/2} S_{n - 1} \int_0^\infty \rho^{k + n - 1} e^{t \rho^2 - \rho^2/2} d\rho = \\ 2^{k/2} (1 - 2 t)^{(-k - n)/2} \left( \frac n 2 \right)_{k/2},$$ where $S_n$ is the surface area of a unit $n$-sphere and $(n)_k$ is the rising factorial.