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%I #18 Sep 27 2017 04:27:46
%S 0,0,0,2,5,11,22,37,61,95,141,203,288,393,531,704,918,1180,1504,1887,
%T 2351,2900,3546,4301,5187,6202,7379,8726,10262,12005,13987,16209,
%U 18716,21521,24652,28135,32013,36291,41028,46244,51977,58262,65155,72667,80872,89798
%N Number of connected undirected unlabeled loopless multigraphs with 4 vertices and n edges.
%C There are 6 basic underlying simple graphs on 4 vertices: the linear chain with 3 edges (a tree), the star graph with 3 edges (a tree), the 4-cycle (quadrangle) with 4 edges, the triangle extended with one edge protruding to a vertex of degree 1 (4 edges), the complete graph on 4 vertices with 6 edges, a graph with 5 edges (removing one from the complete graph).
%H R. J. Mathar, <a href="/A290778/a290778.pdf">Connected multigraphs with 4 vertices (A290778)</a>
%H R. J. Mathar,<a href="http://arxiv.org/abs/1709.09000">Statistics on Small Graphs</a>, arXiv:1709.09000 (2017) Eq. (20).
%H <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (2, 0, 0, -2, -2, 3, 0, 3, -2, -2, 0, 0, 2, -1).
%F G.f.: -x^3*(x^10-x^9-2*x^7+x^6-x^5+3*x^4-x^2-x-2)/( (x-1)^6 *(1+x)^2 *(1+x^2) *(1+x+x^2)^2 ). - _R. J. Mathar_, Aug 11 2017
%e There are a(3) = 2 connected graphs of 3 edges and 4 vertices, the A000055(4) = 2 trees on 4 vertices.
%e There are a(4)=5 connected graphs of 4 edges and 4 vertices: duplicate either the middle or a sided edge of the linear chain, duplicate an edge of the star graph, or take any of the two underlying simple graphs with 4 edges.
%Y Column 4 of A191646.
%K nonn,easy
%O 0,4
%A _R. J. Mathar_, Aug 10 2017
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