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?



Identity for Power Series and Binomial Coefficients
Seems that a general formula for the $x^M$ coefficient of $\exp NF(k,x)$ is
$$
\frac{N}{M} (k^2-k) \left( {(k^2-k) N - (k-1) M - 1 \atop M-1} \right),
$$
which agrees with your formula when $N=M+1$. This should follow from
an explicit formula for $dF(k,x)/dx$ as a degree-$k$ algebraic function of $x$
that's closely related with the inverse function of $y(1-y)^{k-1}$, for which
there's a closed-form power series expansion of $y^\beta$ for all $\beta$;
see for instance 
[these](http://www.math.harvard.edu/~elkies/Misc/catalan.pdf)
[two](http://www.math.harvard.edu/~elkies/Misc/catalan2.pdf)
"one-page papers"
[on my
math webpage](http://www.math.harvard.edu/~elkies/Misc/index.html#papers).


