1. Given an integer $T$ how many $n\times n$ matrices in $\mathbb Z\_{\geq0}^{n\times n}$ we have such that every row and column sum to same $T$?
2. Do the set of constant row and column sum matrices form any kind of algebraic structure (it is definitely definable by linear equations) by which I mean closedness, more than permutation invariance at least for special classes etc.)?



Constant row-column sum matrices?
If you are looking for a simple formula, you are out of luck except for small $n$ or $T$. As Sam mentions in a comment, these are the integer points in a dilated Birkhoff polytope. Since the vertices of the polytope are lattice points, the number of matrices is a polynomial of degree $(n-1)^2$ in $T$ for any fixed $n$ (the Ehrhart polynomial) but the polynomial is only known up to $n=9$.


[This paper](https://arxiv.org/pdf/math/0703600.pdf) and [this paper](https://arxiv.org/pdf/0705.2422.pdf) have some stuff about asymptotics. Searching for these buzzwords will get you more interesting things, like [this paper](https://link.springer.com/content/pdf/10.1007%2Fs10801-008-0155-y.pdf).


