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Number of unlabeled, connected graphs on n vertices whose complements are bipartite.
2

%I #18 Sep 18 2019 04:56:32

%S 1,1,1,2,5,11,32,85,299,1115,5474,32298,251129,2527706,33985846,

%T 611846933,14864650916,488222721984,21712049275189,1308300679611460,

%U 106897965189674281,11852113048215107812,1784730721403509209204,365323537513403184463262

%N Number of unlabeled, connected graphs on n vertices whose complements are bipartite.

%C Equivalently, number of bipartite graphs whose complement is connected. The only bipartite graphs with disconnected complement are complete bipartite graphs. - _Falk Hüffner_, Jan 22 2016

%H Andrew Howroyd, <a href="/A079571/b079571.txt">Table of n, a(n) for n = 0..50</a>

%F a(n) = A033995(n) - floor(n/2).

%t A005142 = Import["https://oeis.org/A005142/b005142.txt", "Table"][[All, 2]];

%t etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n - j], {j, 1, n}]/n]; b];

%t b = etr[A005142[[# + 1]]&];

%t a[n_] := b[n] - Floor[n/2];

%t a /@ Range[0, 50] (* _Jean-François Alcover_, Sep 17 2019 *)

%Y Cf. A005142, A033995.

%K nonn

%O 0,4

%A _Jim Nastos_, Jan 24 2003

%E Corrected and extended using formula by _Falk Hüffner_, Jan 22 2016

%E a(0)=1 prepended and terms a(21) and beyond from _Andrew Howroyd_, Sep 05 2018