Suppose that $\{X_k:k\in\mathbb Z\}$ are strictly stationary random variables such that $\operatorname E|X_0|^3<\infty$. Using the stationarity of the random variables, we can express the second moment of the sum of the random variables $X_1,\ldots,X_n$ in the following way $$ \operatorname E\biggl[\sum_{k=1}^nX_k\biggr]^2=\sum_{k=-(n-1)}^{n-1}(n-|k|)\operatorname E[X_0X_k]. $$
Is there a similar expression for the third moment of the sum of the random variables $X_1,\ldots,X_n$?
We have that $$ \operatorname E\biggl[\sum_{k=1}^nX_k\biggr]^3=\sum_{k=-(n-1)}^{n-1}\sum_{l=-(n-1)}^{n-1}a(n,k,l)\operatorname E[X_0X_kX_l] $$ and I am trying to understand how $a(n,k,l)$ could look like. We can recover a few values of $a(n,k,l)$. For example, $a(n,0,0)=n$, $a(n,-(n-1),l)=0$ for $1\le l\le n-1$. But I am struggling to obtain a general expression.
Any help is much appreciated!