Let $n=2k, k \in Z_+$. Let
$$P_k\left(\frac{t}{\sqrt n}\right)=n!\sum_{\begin{smallmatrix} n_1+\ldots+n_k=n \\ j=n_2+2n_3+\ldots+(k-1)n_k\end{smallmatrix}}\left(\frac{-t^2}{n}\right)\frac{1}{\prod_{i=1}^k((2i-1)!)^{n_i}n_i!}
$$
Expand $$ \left[e^{-t^2/n}P_k \left(\frac{t}{\sqrt n}\right)\right]^n $$ in powers of $\displaystyle{\frac{t}{\sqrt n}}$ to get all terms of the form powers of $\displaystyle{\frac{t}{ n^j}}$, $j=0,1,\ldots, m$ (here $k=m+1$). Go up to $\displaystyle{\left(\frac{t}{\sqrt n}\right)^{4m}}$.
Thank you.