Let $Vir_c$ denote the state space of the Virasoro VOA with central charge $c$. Let $n$ be a positive integer(i.e. $\geq 0$) and let $(n_k,...,n_2)$ be a sequence of positive integers. I am interested in knowing the explicit formula of the form $$L_{(n)}:L^{n_k}_{-k}L^{n_{k-1}}_{-(k-1)}...L^{n_2}_{-2}:=\sum_{s\geq 2,i_k\geq 0,..,i_2\geq 0}C(n;n_k,...,n_2;i_s,...,i_2):L^{i_s}_{-s}L^{i_{s-1}}_{-(s-1)}...L^{i_2}_{-2}:,$$ where $C(n;i_1,...,i_k)$ are some coefficients that depend on the prescribed indices and $:ab:$ denotes the normal order of states $a$ and $b$. Essentially these are structure constants of the OPE algebra. I have derived a few special cases, which has been rather painful, and firmly believe such a formula must exist and should be known. I have 2 questions,
Can this community help me with finding a reference for such a formula, with explicitly prescribed structure constants?
Just as a curiosity and as a natural generalization of the above, for $n\in\mathbb{Z}$ and $(m_l,...,m_2)$ a sequence of positive integers, is the explicit formula of the form $$:L^{r_l}_{-l}...L^{r_2}_{-2}:_{(n)}:L^{n_k}_{-k}...L^{n_2}_{-2}:=\sum_{s\geq 2,i_k\geq 0,..,i_2\geq 0}C(n;n_k,...,n_2;m_l,...,m_2;i_s,...,i_2):L^{i_s}_{-s}L^{i_{s-1}}_{-(s-1)}...L^{i_2}_{-2}:$$ known?
Thank you.