Red-headed birds

We live in north Louisiana and have seen four large birds (12 to 16 inches) that resemble the red-headed woodpecker. They are jet black, a red head with white showing on the wings while in flight but not noticeable when on a limb. They seem to love, what we call, a Chinese tallow tree. We saw one about a year ago but now we have four. What are they and are they migrating through this area this time of the year? They stand out because of their size. They are about the size of a crow, except a little thicker in the body area. We are very interested to know who our new comers are.