Let $X\in\{-1,+1\}$ be a binary random variable. Then for the indicator function $\mathbf{1}_{X=x}$, we have $$\mathbf{1}_{X=x}=\frac{X-1}{-2}\frac{x-1}{-2}+\frac{X+1}{2}\frac{x+1}{2}=\frac{1+xX}{2},$$ for $(x,X)\in\{-1,+1\}^2$. The expression above gives $$\mathbb{P}(X=x)=\mathbb{E}[\mathbf{1}_{X=x}]=\frac{1+x\mathbb{E}[X]}{2}$$ or for a joint pmf $$\mathbb{P}(X_1=x_1,X_2=x_2)=\mathbb{E}[\mathbf{1}_{X_1=x_1}\mathbf{1}_{X_2=x_2}]=\frac{1+x_1\mathbb{E}[X_1]+x_2\mathbb{E}[X_2]+x_1 x_2 \mathbb{E}[X_1 X_2]}{4}.$$ Based on this we can find an expression of the joint pmf for $N$ binary random variables parametrized over the moments $\mathbb{E}[X_1 X_2\ldots X_i]$.
However, if we include one more point to the support lets say $0$, we have $X\in\{-1, 0,+1\}$ and $$\mathbf{1}_{X=x}=\frac{Xx(x-1)(X-1)}{4}+(x-1)(X-1)(x+1)(X+1)+\frac{Xx(x+1)(X+1)}{4}$$ and the evaluation of $\mathbb{E}[\mathbf{1}_{X=x}]$ is already messy, which makes hard to find the expression of joint pmfs. Is there any other way to find the formula of a pmf parametrized over the moments similarily to the binary case?