找出 n 个节点的所有可能的连通图和有向图的数量

Question

Hello stackoverflow community,
I need to find out the number of all possible connected and directed graphs for n nodes.

For example: 3 node graphs can have 13 possible combinations which are:

CONDITIONS: As you can see in the above image,
->3 node connected graph can never have just 1 edge, atleast two edges are required to connect all 3 nodes. So all nodes should be connected.
->Maximum edges = 6 in 3 nodes. (See graph number 13 in the image, it got 6 edges)
->There can be no self edge.

Similarly 4 nodes will have 199 connected directed graphs.
To sum up:
2 nodes = 3 graphs
3 nodes = 13 graphs
4 nodes = 199 graphs
5 nodes = 9364 graphs
6 nodes = 1530843 graphs

I need a formula for F(n) so that I can get the total number of graphs for n nodes just by calculating the formula instead of doing exhaustive search to try each and every possible combination.
F(2) = 3
F(3) = 13
F(4) = 199
F(5) = 9364
F(6) = 1530843

what is F(n) where n can be any natural number ?

I have been trying to solve this puzzle since many days but couldn't figure it out, so I am using exhaustive methods to find out the number, but they are not feasable.

Answer 1

The Online Encyclopedia of Integer Sequences (OEIS) is useful for things like this. Below is a link to this sequence, which in turn has references you can use to learn more.

http://oeis.org/A003085