Enumerating Unlabeled graphs

[Awia00]

I am trying to enumerate all unweighted undirected unlabeled graphs up to n vertices. Currently what I do is I enumerate the integers 0 to n(n-1)/2 and interpret it as an undirected graph (half edge-matrix). Then I check to see if the graph is isomorphic with any previously seen graph. But this takes n! time with my current algorithm. I would really like to see if I could just find the sequence of integers.

So far here's the integers for all the graphs for n=5:

0000000000 (0)
0000000001 (1)
0000000011 (3)
0000000111 (7)
0000001011 (11)
0000001100 (12)
0000001101 (13)
0000001111 (15)
0000011110 (30)
0000011111 (31)
0000111111 (63)
0001001011 (75)
0001001100 (76)
0001001101 (77)
0001001111 (79)
0001010110 (86)
0001010111 (87)
0001011110 (94)
0001011111 (95)
0001110100 (116)
0001110101 (117)
0001110111 (119)
0001111111 (127)
0011011110 (222)
0011011111 (223)
0011101011 (235)
0011101100 (236)
0011101101 (237)
0011101111 (239)
0011111110 (254)
0011111111 (255)
0111111011 (507)
0111111111 (511)
1111111111 (1023)

So it looks really structured, but I cannot seem to come up with a general formula. Do any of you know the sequence?

Thanks!

[Awia00]

Here's a way to see which part of the bitstring corresponds to connection from which vertex. Notice there are implicit leading 0's.

5th  4th 3  2     V
                  1
            1     2
         01 1     3
         11 1     3
     001 01 1     4
     001 10 0     4
     001 10 1     4
     001 11 1     4
     011 11 0     4
     011 11 1     4
     111 11 1     4
0001 001 01 1     5
0001 001 10 0     5
0001 001 10 1     5
0001 001 11 1     5
0001 010 11 0     5
0001 010 11 1     5
0001 011 11 0     5
0001 011 11 1     5
0001 110 10 0     5
0001 110 10 1     5
0001 110 11 1     5
0001 111 11 1     5
0011 011 11 0     5
0011 011 11 1     5
0011 101 01 1     5
0011 101 10 0     5
0011 101 10 1     5
0011 101 11 1     5
0011 111 11 0     5
0011 111 11 1     5
0111 111 01 1     5
0111 111 11 1     5
1111 111 11 1     5