I have the problem for computing the j-derivative of a logarithm, with $j\gg1$
\begin{equation}
c_j=\left.\frac{\partial^j}{\partial s^j}\log\left(1+Ae^s+Be^{2s}\right)\right|_{s=0},
\end{equation}
being A and B real numbers ($0<A,B<1$). What I have done is to perform a Taylor expansion of the logarithm, finding
\begin{equation}
c_j=\left.\frac{\partial^j}{\partial s^j}\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}e^{ns}\left(A+Be^s\right)^n\right|_{s=0}.
\end{equation}
Finally, I can use the Newton binomia expression and derive inside the sum
\begin{equation}
c_j=\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}\sum_{k=0}^{n}{n\choose k}(2n-k)^jA^{n-k} B^{k}
\end{equation}
My question is to approximate this expression in the limit when $j\gg1$. I though about using some Stirling kind approximation, but I did not find any solution. Thanks for any help