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%I #16 Sep 21 2019 14:40:45
%S 0,0,1,6,42,402,5381,112776,3935471,240684836,26449057257,
%T 5289513580458,1939502108505917,1311274498490104492,
%U 1642800188822966309834,3831285832174735713684706,16703340559932677463553709189,136661710199022168890320488632600,2105815888079982128884579271408161673,61310553163194788144046000967760340771668
%N Number of simple connected graphs with n nodes rooted at one non-edge.
%H Jean-François Alcover, <a href="/A304071/b304071.txt">Table of n, a(n) for n = 1..40</a>
%F a(n) + A303830(n) = A303831(n).
%e a(3)=1: the non-edge joins the two leaves. a(4)=6: quadrangle: the non-edge is a diagonal; triangle with protruding edge: the non-edge joins the leaf with a node of degree 2; quadrangle with diagonal: the non-edge is the other diagonal; tetrahedron: no contribution; linear chain: the non-edge either joins the two leaves or a leaf with a node at distance 2; star graph: the non-edge joins two leaves.
%t A303830 = Import["https://oeis.org/A303830/b303830.txt", "Table"][[All, 2]];
%t A303831 = Import["https://oeis.org/A303831/b303831.txt", "Table"][[All, 2]];
%t a[n_] := A303831[[n]] - A303830[[n]];
%t a /@ Range[1, 40] (* _Jean-François Alcover_, Sep 21 2019 *)
%Y Cf. A001349 (not rooted), A126122 (not necessarily connected)
%K nonn
%O 1,4
%A _Brendan McKay_, May 05 2018
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