I was reading D.Knuth's book "Concrete mathematics. Foundation of computer science". In chapter 8.5 Hashing was given: $$B_n(z) = \sum_{n_1,n_2,...,n_m \geqslant \ 0 \\ n_1+n_2+...n_m = \ n } \begin{pmatrix} n \\ n_1,n_2, ..., n_m \end{pmatrix} z^{\begin{pmatrix} n_1 \\2 \end{pmatrix} + \begin{pmatrix} n_2 \\2 \end{pmatrix} +...+\begin{pmatrix} n_m \\2 \end{pmatrix} m^{-n} } $$
We can recognize it is m-fold convolution. Indeed, if we consider the exponential super-generating function
$$G(w,z) = \sum_{n=0} B_n(z) \frac {m^n w^n}{n!} $$
we can readily verify that $G(w,z)$ is simply an m-th power:
$$G(w,z) = \left( \sum_{k \geqslant 0} z^{\begin{pmatrix} k \\2 \end{pmatrix}} \frac {w^k}{k!} \right)^m $$
Take a derivative of $G(w,z)$
$$\frac d{dz}G(w,z) = \sum_{\geqslant \ 0} B_n' \frac {m^n w^n}{n!} = m\left( \sum_{k \geqslant \ 0} z^{\begin{pmatrix} k \\2 \end{pmatrix}} \frac {w^k}{k!} \right)^{m-1} \sum_{k \geqslant \ 0} \begin{pmatrix} k \\2 \end{pmatrix} z^{\begin{pmatrix} k \\2 \end{pmatrix} -1} \frac {w^k}{k!}; $$
The question is: What is $\sum_{k \geqslant \ 0} \begin{pmatrix} k \\2 \end{pmatrix} z^{\begin{pmatrix} k \\2 \end{pmatrix} -1} \frac {w^k}{k!}$ and how it appeared?