Simply put, have got is actually the proper/grammatical way to say it. This is the type of grammar you will most likely see in books, news articles and formal speech. got with the dropped have is the same thing except that it's just a more colloquial way of saying it mostly heard only in daily English. For example, if you go to North America and listen to how people use this expression, you will see that most of them would just say I got instead of I've got or I have got. But that, of course, depends on who you're speaking with. And that's pretty much all that can be said about this.
How many children do you have?
I got one girl and two boys.