What is the term used to refer multi-dimensional array of programming languages in mathematics?
I thought it was tensor, but I learned that tensors should obey some restrictions(indepentent on coordinate transform, ...)
For example:
A order 0 tensor is a scalar.
A order 1 tensor is a vector.
A order 2 tensor is a matrix.
What do you call order 3 tensor-like something but doesn't have to be independent on coordinate transformation?
My guess is that you are misunderstanding what it means to be independent under coordinate transformations and you can call your object a rank 3 tensor, or better yet, an element of $\mathbb{R}^{n_1\times n_2 \times n_3}$.
For example, invariance under coordinate transformations does not mean that your array has to be symmetric in any way.
Invariance under coordinate transformations only makes sense in a physical context and if your array of numbers actually comes from such a context then chances are that your array is invariant without you knowing it. Without going into details, any list or array of numbers can represent a tensor (-field, to be precise) but that depends on where they come from in your context and I don't think this discussion has anything to do with what you want to do.