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%I #33 Mar 21 2023 05:29:16
%S 1,3,8,30,118,548,2790,16029,101353,706572,5375249,44402094,395734706,
%T 3786401086,38711834576,421217184135,4860174299186,59278045511959,
%U 762055884150141,10299293881159294,145994591873294780,2165938721141964179,33564939201581495090,542344644703485899950,9122110321170144880053
%N The number of unlabeled connected fairly 4-regular multigraphs of order n, loops allowed.
%C Edges are undirected, vertices not labeled. "Fairly" means that each vertex has degree 4, but two of these edges do not connect to a second vertex; they are "fins" in CAD speak or "half-edges" in perturbation theory. The two fins may be attached to the same or to two different nodes. In the usual mathematical nomenclature these are connected graphs of order n+2 with two vertices of degree 1 and n vertices of degree 4, loops allowed.
%H H. Kleinert, A. Pelster, B. Kastening, and M. Bachmann, <a href="https://doi.org/10.1103/PhysRevE.62.1537">Recursive graphical construction of Feynman diagrams and their multiplicities in Phi^4 and Phi^2*A theory</a>, Phys. Rev. E 62 (2) (2000), 1537 Table II.
%H R. J. Mathar, <a href="/A361135/a361135.pdf">Illustrations</a>
%Y Cf. A085549 (4-regular), A352174 (assuming rooted external legs).
%K nonn,hard
%O 1,2
%A _R. J. Mathar_, Mar 02 2023
%E Terms a(7) and beyond from _Andrew Howroyd_, Mar 05 2023