Consider the multi-index $\gamma=(\gamma_1,\ldots, \gamma_n)$. Is there a closed form for the sum $\sum_{|\gamma|=k} \gamma!$ in terms of $n$ and $k$? Asymptotics, or good upper bounds are also very helpful.

Here is what I have tried. Let $f(x)=\sum_{i=0} i! x^{i+2}$. This generating function satisfies the ODE $x^2f'(x)=f(x)-x$. Then the sum $\sum_{|\gamma|=k} \gamma!$ is the coefficient of $x^{k+2n}$ in $f(x)^n$. However I don't have any means to find this coefficient.