A133330 implicit in "On sums of figurate numbers by using techniques of poset representation theory"

[Richard Mathar]

Neil,

Brendan and I have agreed on that the sequence A126757 enumerates
*connected* graphs, not biconnected ones. Therefore, I suggest the
following update to A126757 (that extends and makes it a counterpart
of A006787; please notice that the value of a(1) is corrected):

%S A126757 1, 1, 1, 2, 4, 8, 18, 47, 137, 464, 1793, 8167, 43645,
275480, 2045279, 17772647, 179593823
%N A126757 Number of n-node connected graphs with no cycles of length
less than 5.

Under this definition, Vladeta's conjecture becomes an easily established fact.

Also, since a number of adjacent sequences mentions word "biconnected"
as well, it needs to be checked if they really mean biconnected, not
just connected:
http://public.research.att.com/~njas/sequences/A126749
http://public.research.att.com/~njas/sequences/A126750
http://public.research.att.com/~njas/sequences/A126751
http://public.research.att.com/~njas/sequences/A126752
http://public.research.att.com/~njas/sequences/A126753
http://public.research.att.com/~njas/sequences/A126754
http://public.research.att.com/~njas/sequences/A126755
http://public.research.att.com/~njas/sequences/A126756
http://public.research.att.com/~njas/sequences/A126758

Regards,
Max




vj> From seqfan-owner at ext.jussieu.fr  Tue Jun 17 02:56:03 2008
vj> From: "Vladeta Jovovic" <vladeta at eunet.yu>
vj> To: <seqfan at ext.jussieu.fr>
vj> Subject: A conjecture
vj> Date: Tue, 17 Jun 2008 02:52:00 +0200
vj> 
vj> > Subject: COMMENT FROM Vladeta Jovovic RE A126757
vj> >
vj> >
vj> > %I A126757
vj> > %S A126757 0, 1, 1, 2, 4, 8, 18, 47, 137, 464, 1793, 8167, 43645, 275480
vj> > %N A126757 Number of biconnected 4-cycle-free graphs on n nodes with 
vj> > clique number 2.
vj> > %C A126757 Conjecture: Inverse Euler transform of A006787.
vj> > %O A126757 1
vj> > %K A126757 ,nonn,
vj> > %A A126757 Vladeta Jovovic (vladeta at Eunet.yu), Jun 16 2008

If this is correct we can generate more terms in A126757 by exploiting the full
list of values in A006787 with the help of the EULERi Maple implementation:

1, 1, 1, 2, 4, 8, 18, 47, 137, 464, 1793, 8167, 43645, 275480, 2045279,
17772647, 179593823, 2098423758, 28215583324

Note that this generates a 1, not a zero in front.

Richard