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\begin{document}
With the above notations,
\begin{enumerate}
\item $[W_0^k, \dots, W_n^k]_{(C_1, \dots, C_n)} = 0$ provided $n >k$ ;
\item $\begin{multlined}[t]
[W_0^k, \dots, W_n^k]_{(C_1, \dots, C_n)} = \sum_{j=0}^{k-1}[W_0^{k-j-1}, \dots, W_{n-1}^{k-j-1}]_{(C_1, \dots,C_{n-1})}C_nW_n^j \\ = \sum_{j=0}^{k-n}[W_0^{k-j-1}, \dots, W_{n-1}^{k-j-1}]_{(C_1, \dots,C_{n-1})}C_nW_n^j
\end{multlined}$
\item $\begin{aligned}[t]
[W_0^k, \dots, W_n^k]_{(C_1, \dots, C_n)} &= \sum_{j=0}^{k-1}[W_0^{k-j-1}, \dots, W_{n-1}^{k-j-1}]_{(C_1, \dots,C_{n-1})}C_nW_n^j \\ &= \sum_{j=0}^{k-n}[W_0^{k-j-1}, \dots, W_{n-1}^{k-j-1}]_{(C_1, \dots,C_{n-1})}C_nW_n^j
\end{aligned}$
\item \begin{align*}
\SwapAboveDisplaySkip %fourni par mathtools
[W_0^k, \dots, W_n^k]_{(C_1, \dots, C_n)} = \sum_{j=0}^{k-1}[W_0^{k-j-1}, \dots, W_{n-1}^{k-j-1}]_{(C_1, \dots,C_{n-1})}C_nW_n^j \\ = \sum_{j=0}^{k-n}[W_0^{k-j-1}, \dots, W_{n-1}^{k-j-1}]_{(C_1, \dots,C_{n-1})}C_nW_n^j
\end{align*}
\end{enumerate}
\end{document}