For the Coursera offerings, any order/sequence thoughts?
No, you can take the deep learning specialization without having taken the ML course. However, the specialization doesn't cover some of the material in the ML course. That may be a reason to take the ML course. But the ML course is not required as a "conceptual and mathematical foundation" for the specialization. The specialization is largely self-contained. Within the specialization, you are better off taking the 5 courses in the same order they are given.
Hello everyone, I accomplished the deep learning specialization before accomplishing the machine learning course, I didn't face problems to understand the materials of DL specialization, knowing that my academic background is in Mathematics.
I prefer and recommend others (especially who are not familiar with mathematics) to take the ML course before taking the DL specialization, the DL is more advanced than the ML course, indeed you can start with the DL specialization, it's self-contained as tmatha mentioned.