$n^2$ 求 $\ln$

\[\texttt{Given }F,\ G=\ln F\\ \exp G=F\\ (\exp G)'=F'\\ G'F=F'\\ [x^k]G'F=[x^k]F'\\ \sum_{i=0}^k(i+1)g_{i+1}f_{k-i}=(k+1)f_{k+1}\\ (k+1)g_{k+1}f_0=(k+1)f_{k+1}-\sum_{i=0}^{k-1}(i+1)g_{i+1}f_{k-i}\\ g_{k+1}=f_{k+1}-{1\over k+1}\sum_{i=0}^{k-1}(i+1)g_{i+1}f_{k-i} \]