Q:

Expressing $\mathbb{E}[X_1X_2 \ldots X_n]$ as their pairwise correlations

Let's say we have a sequence of almost surely bounded random variables $X_1,\ldots,X_n$, and their pairwise correlation is given by some covariance structure
$$ \mathbb{E}[X_i X_j] = K_{ij}. $$
Is it possible now to express $\mathbb{E}[X_1 X_2\ldots X_n]$ in terms of $(K_{ij})_{i,j}$? Preferably I'd like to say something along the lines of
$$ \mathbb{E}[X_1 X_2 \ldots X_k] = K_{12} K_{23} \ldots K_{k-1k}, $$
but I don't even know if this is possible! I can't seem to find anything on expectations of products involving multiple correlated random variables.
EDIT: If we restrict ourselves to the case where the $X_i$ are symmetric random variables, then $X_i \stackrel{d}{=} -X_i$, so for odd $k$ we can at least say:
$$ \mathbb{E}[X_1X_2\ldots X_k] = (-1)^k \mathbb{E}[X_1X_2\ldots X_k], $$
implying that $\mathbb{E}[X_1 X_2 \ldots X_k] = 0$.

A:

Example 1: Let $X_1,X_2,X_3$ are i.i.d. $\text{Bernoulli}(\tfrac{1}{2})$ random variables. 
Then, $K_{ij} = \mathbb{E}[X_iX_j] = \begin{cases}\tfrac{1}{2} & \text{if} \ i = j\\ \tfrac{1}{4} & \text{if} \ i \neq j\end{cases}$, and $\mathbb{E}[X_1X_2X_3] = \tfrac{1}{8}$. 
Example 2: Let $X_1,X_2,X_3$ be random variables such that $(X_1,X_2,X_3)$ takes on the values $(0,0,0), (0,1,1), (1,0,1), (1,1,0)$ each with probability $\tfrac{1}{4}$. 
Then, $K_{ij} = \mathbb{E}[X_iX_j] = \begin{cases}\tfrac{1}{2} & \text{if} \ i = j\\ \tfrac{1}{4} & \text{if} \ i \neq j\end{cases}$ (just as in Example 1), but $\mathbb{E}[X_1X_2X_3] = 0$. 
So in general, you need more information than just the covariances $K_{ij} = \mathbb{E}[X_iX_j]$ to compute $\mathbb{E}[X_1X_2 \cdots X_n]$.

