For given $k$, we know that \begin{align*} \sum_{\tau = 1}^{k-1}(-1)^{\tau} \binom{k-1}{\tau} \sum_{t_1 + \cdots + t_{\tau}=k, t_j\geq 1}\beta_{t_1}\beta_{t_2}\cdots\beta_{t_{\tau}} \end{align*} is equal to $\frac{1}{k!}\frac{d^k}{d \alpha^k} (1-g(\alpha))^{k-1}\bigg|_{\alpha = 0}$, with $g(\alpha) = \sum_{k\geq 1}\beta_k\alpha^k$. Basically we first choose $\tau$ groups that contribute $\alpha$. And sum over all possible $\tau$ from $1$ to $k-1$.
Now if I am interested in \begin{align*} \sum_{\tau = 1}^{k-2}(-1)^{\tau} \binom{k-1}{\tau+1} \sum_{t_1 + \cdots + t_{\tau} = k-1, t_j\geq 1}\beta_{t_1}\beta_{t_2}\cdots\beta_{t_{\tau}}. \end{align*} Here we have $\tau+1$ groups but only $\tau$ of them contribute to $\alpha$. I would like to know the corresponding generating function for this sum. Any hint would be appreciated.
I think it would be helpful to write $\frac{1}{\tau+1} = \int_{0}^1x^\tau dx$. Then we can assign each $x$ to the coefficients $\beta_{t_1},\cdots, \beta_{t_{\tau}}$. And this should be related to the function $1-g(\alpha)\cdot x$, followed by integration wrt $x$.