A327074 - OEIS

%I #8 Sep 23 2019 09:45:39

%S 0,0,1,0,1,4,25,197,2454,48201,1604016,93315450,9696046452,

%T 1822564897453,625839625866540,395787709599238772,

%U 464137745175250610865,1015091996575508453655611,4160447945769725861550193834,32088553211819016484736085677320,467409605282347770524641700949750858

%N Number of unlabeled connected graphs with n vertices and exactly one bridge.

%C A bridge is an edge that, if removed without removing any incident vertices, disconnects the graph. Unlabeled graphs with no bridges are counted by A007146 (unlabeled graphs with spanning edge-connectivity >= 2).

%H Gus Wiseman, <a href="/A327074/a327074.png">The a(5) = 4 unlabeled connected graphs with n vertices and exactly one bridge.</a>

%F G.f.: (f(x)^2 + f(x^2))/2 where f(x) is the g.f. of A007145. - _Andrew Howroyd_, Aug 25 2019

%t A007145 = Cases[Import["https://oeis.org/A007145/b007145.txt", "Table"], {_, _}][[All, 2]];

%t f[x_] = A007145 . x^Range[lg = Length[A007145]];

%t (f[x]^2 + f[x^2])/2 + O[x]^lg // CoefficientList[#, x]& (* _Jean-François Alcover_, Sep 23 2019 *)

%Y The labeled version is A327073.

%Y Unlabeled graphs with at least one bridge are A052446.

%Y The enumeration of unlabeled connected graphs by number of bridges is A327077.

%Y BII-numbers of set-systems with spanning edge-connectivity >= 2 are A327109.

%Y Cf. A001349, A002494, A007146, A263296, A327075, A327111, A327144, A327145.

%K nonn

%O 0,6

%A _Gus Wiseman_, Aug 24 2019

%E Terms a(6) and beyond from _Andrew Howroyd_, Aug 25 2019