Construct Degree

You are given an integer $N$. Count the number of distinct arrays $D$ of size $N$ containing non-negative integers such that it is possible to construct a graph $G$ of $N$ nodes with the following constraints: 1. For each $i$ ($1 \leq i \leq N$), node $i$ has degree $D_i$ in $G$. 2. $D_1 + D_2 + \ldots + D_N = 2N$. 3. $G$ is [simple](https://mathworld.wolfram.com/SimpleGraph.html), [connected](h