This question concerns a combinatorial identity obeyed by power series coefficients. Throughout we let $[x^{M}]\{\phi(x)\}$ denote the coefficient of $x^{M}$ in a power series $\phi(x)$.

Let $k$ be a positive integer, and consider the function $F(k,x)$ defined as the following power series in $x$:

\begin{equation} F(k,x)=\sum_{s=1}^{\infty} \frac{(-1)^{s-1}}{s^{2}}\binom{s \ k}{s+1}(s+1)\ x^{s}. \end{equation}

I am interested in the series coefficients of the function $\exp(N F (k,x))$ for positive integer $N.$

Through comparison of various formulas that arose in a research project, I have been lead to the following identity for the case $N=M+1$:

\begin{equation} [x^{M}]\{e^{(M+1)F(k,x)}\}= \frac{k(M+1)}{k+(k-1)M}\binom{(k-1)^{2}M+k(k-1)}{M}~. \end{equation}

Although I am convinced that this identity is true, I have no idea how to demonstrate it, nor do I have any idea why this power series coefficient has such a simple expression. Thus, my main question is how can this identity be motivated and proven ?

More generally, can we determine the coefficient $[x^{M}]\{e^{N F(k,x)}\}?$

I am also interested in a generalization which depends on an additional positive integer $j$. Specifically, set

\begin{equation}
F(k,j,x)=\sum_{s=1}^{\infty} \frac{(-1)^{s-1}}{s^{2}}\binom{s \ k}{s\ j+1}(s\ j+1)\ x^{s}~.
\end{equation}
The previous function is recovered for the special case $j=1.$

Can the coefficients $[x^{M}]\{e^{NF(k,j,x)}\}$ be similarly determined?