There is no nice formula, I’m afraid. What you want is the number of simple graphs on $n$ unlabelled vertices. For questions like this the On-Line Encyclopedia of Integer Sequences can be very helpful. I searched in on the words
unlabeled graphs, and the very first entry returned was OEIS A000088, whose header is Number of graphs on n unlabeled nodes. A quick check of the smaller numbers verifies that graphs here means simple graphs, so this is exactly what you want. It tells you that your $1,2$, and $4$ are correct, and that there are $11$ simple graphs on $4$ vertices. You should check your list to see where you’ve drawn the same graph in two different ways. If you get stuck, this picture shows all of the non-isomorphic simple graphs on $1,2,3$, or $4$ nodes. The OEIS entry also tells you how many you should get for $5$ vertices, though I can’t at the moment point you at a picture for a final check of whatever you come up with.