[seqfan] Re: Need help with graph sequences

[Max Alekseyev]

On Sat, Feb 21, 2009 at 10:39 AM, Tanya Khovanova
<mathoflove-seqfan at yahoo.com> wrote:
>
> Hello SeqFans,
>
> I just submitted two sequences below. My inefficient program counts only the first 6 terms. The problem is that the second sequence exists in the OEIS, but I do not have any good reasons to believe that it is the same as mine, I need the next term to make sure. Can you do that?
>
> %S A157015 1,2,3,8,18,60
> %N A157015 Number of graphs with n vertices, such that a bipartite connected component exists.
> %C A157015 There exists a vertex which doesn't belong to a cycle of odd length.
>
>
> %S A157016 0,0,1,3,16,96
> %N A157016 Number of graphs with n vertices such that a bipartite connected component doesn't exist.
> %C A157016 Any vertex is connected to a cycle of odd length.
> %t A157016 cbs[g_] := Or @@ Map[BipartiteQ, Map[InduceSubgraph[g, #] &, ConnectedComponents[g]]] Table[Count[Map[cbs, ListGraphs[n]], False], {n, 6}]

A157016 is the Inverse Euler of A001349(n)-A005142(n)

1, 0, 0, 1, 3, 16, 96, 812, 10960, 260494, 11713892, 1006689874,
164059932991, 50335918374390, 29003488479342757, 31397381309486943966,
63969560164056605070568, 245871831711240782889155306,
1787331725280384281423635294536, 24636021429463931875328536155116965

A157015(n) = A000088(n) - A157016(n):

0, 1, 2, 3, 8, 18, 60, 232, 1386, 14174, 291276, 12307990, 1031239601,
166112993562, 50667177892731, 29104660317364802, 31455540470952824360,
64032442292149794564470, 245999865227419124242896312,
1787823661072649054471336315803

Regards,
Max