Expansion of Elementary Symmetric Functions

Question

Let $$e_k(X_1,X_2,\dots, X_n)=\sum_{1\le i_1<i_2<\dots<i_k\le n}X_{i_1}X_{i_2}\dots X_{i_k}$$ be the elementary symmetric function of degree $k$ and $$p_k(X_1,X_2,\dots, X_n)=\sum_{1\le i\le n}X_{i}^k$$ be the power sum symmetric function of degree $k$. Is there any way to write $$e_1^n=\sum_{i,j=1}^{n}(a_i\cdot e_i +b_j\cdot p_j +c_{i,j}\cdot e_i \cdot p_j)$$ That is, to write $e_1^n$ as the sum of various $e_i$'s, $p_j$'s, and their products. I am also interested in any well known formulae for $e_1^n$, even if they don't match the form above. Similar formulae would be Newton's identities, one of which being $$e_k=\frac{1}{k}\sum_{i=1}^k (-1)^{i-1}e_{k-i}\cdot p_i$$ Testing this out we have $$\begin{gather} e_1(X_1)=e_1 \\ e_1(X_1,X_2)^2=p_2+2e_2 \\ e_1(X_1,X_2,X_3)^3=e_1p_2+2e_2p_1 \\ \vdots \end{gather}$$