Let $X\sim \mathrm{Bin}(p_x,1)$ , $Y\sim \mathrm{Bin}(p_x,1)$ and $V\sim \mathrm{Bin}(p_v,1)$ be three binomial random variables, that are NOT independent from each other. Note that $X$ and $Y$ are sampled from the same binomial distribution $\mathrm{Bin}(p_x,1)$. My objective is to find an expression for the covariance \begin{align}\operatorname{Cov}(XY,V)&\end{align}
From https://stats.stackexchange.com/questions/389662/covariance-of-products-of-dependent-random-variables and Eq. 12 in https://www.jstor.org/stable/2286081 (On the Exact Covariance of Products of Random Variables), I learned that this covariance can be expressed as: \begin{align}\operatorname{Cov}(XY,V)&=\operatorname{E}(X)\operatorname{Cov}(Y,V) + \operatorname{E}(Y)\operatorname{Cov}(X,V) + \operatorname{E}[(\Delta_Y)(\Delta_X)(\Delta_V)]&\end{align}
where $\Delta_X = X - \mathbb E[X]$, $\Delta_Y = Y - \mathbb E[Y]$ and $\Delta_Z = Z - \mathbb E[Z]$. Note that if the three random variables are normally distributed, the third moment disappear (Anderson, 1958). So I do wonder what happen in the case of binomial random variables. Also, keep in mind that this variables can be normalised as to obtain mean 0 and variance 1.
Similarly, I am interested in finding an expression for:
\begin{align}\operatorname{Cov}(X^2,V)&\end{align}
Anderson, T. W. An Introduction to Multivariate Statistical Analysis. New York: John Wiley and Sons, 1958.