Suppose $X \sim Bin(n,p)$ and $Y \sim Bin(n,1-p)$, How is $X+Y$ distributed? I know that for independent variables one can do the same as:

Sum of two independent binomial variables

Furthermore i have seen this post:

Addition of two Binomial Distribution

However $X$ and $Y$ do not necessarily need to be independent of each other.

Backstory: I am trying to calculate the entropy of some $Z = X_1 + X_2 + \dots + X_n$ where each $X_i$ is either $Bin(n,p)$ or $Bin(n,1-p)$ depending on its parent node.

For example lets take a graph that has only outgoing edges of degree 2 beginning from some source $X_0 \sim (\frac{1}{2},\frac{1}{2})$. If we compare layer $2$ and layer $3$, we sent $2^3$ nodes to $2^4$ nodes. The probability for a set of child nodes to get certain states is $Bin(n,p)$ when the parent has state $1$ and $Bin(n,1-p)$ if the parent has state $-1$. We proceed this until we reached some threshold layer $d$.

What is the probability distribution of $Z = \sum X_i^{(d)}$