Actually, it took me four days to figure it out. You are right. The explanations regarding those equations somehow are not very clearly for me, a novice, which may be easy to understand for higher-leveled guys.
It seems that I understand the coefficient on the right-hand side. What we want to do is get the coefficient with particle 1 in a certain state E1, particle 2 in a certain state E2 , ..., particle N in a certain state EN, i.e. C(E1, E2, ..., EN, t), which corresponds to $\bar{C}(n_1, n_2, ...
On page 9 of *Quantum theory of many-particle systems* by Alexander L. Fetter and John Dirk Walecka, during the derivation of the second-quantised kinetic term, there is an equality equation below:
>\begin{align}
\sum_{k=1}^{N} \sum_{W} & \langle E_k|T|W\rangle C(E_1, ..., E_{k-1}, W...