In my research (coming from computer science), I have encountered a family of discrete probability distributions that seems to be some sort of generalization of the binomial distribution. A distribution in this family has a parameter $n \in \mathbb{N}$ and parameters $p_{0},... ,p_{n-1} \in [0,1]$. The distribution has $n+1$ possible values. Denote the respective probablities $q_0,...,q_n$. So far, I have managed to obtain the descriptions for the first few $n$, which are the following:

- If $n=1$, the distribution is: \begin{align*}q_0 =& 1-p_0\\q_1 =& 1-q_0\end{align*}
- If $n=2$, the distribution is: \begin{align*}q_0=&(p_0-1)^2\\ q_1=&(p_0-1)p_0(p_1-1)\\ q_2 =& 1-q_0-q_1\end{align*}
- If $n=3$, the distribution is: \begin{align*}q_0=&(1 - p_0)^3\\q_1 =& 3 (p_0-1)^2 p_0 (p_1-1)^2 \\ q_2=& -3 (p_0-1) p_0 (p_1-1)^2 (2 (p_0-1) p_1-p_0) (p_2-1)\\q_3 =& 1-q_0-q_1-q_2 \end{align*}

I could give the descriptions for few higher $n$, but it becomes lots of symbols quickly.

I have noticed, that when I set $p_i = 0 \; \forall i = 1,...,n-1$ it degenerates to the distribution $Bin(n,p_0)$. I have several goals with this. First, I wanted to ask if this resembles some well known distribution family generalizing the binomial distribution? Can anyone suggest possible general formulas that would produce $q_i$ for a given $n$? Assuming, we have a generalization, what would be the measure of all the distributions representable as a member of this family for given $n$ or some upper bound of such measure? Denoting this measure $l_n$, I have computed that: $l_1 = \sqrt{2}, l_2=\frac{1}{\sqrt{3}},l_3 = \frac{27}{280},l_4=\frac{73162}{5010005\sqrt{5}}$.