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%I #11 Mar 21 2023 23:08:38
%S 1,2,9,49,338,2744,26025,282419,3463502,47439030,718618117,
%T 11937743088,215896959624,4224096594516,88919920910684,
%U 2004237153640098,48165411560792500,1229462431057436457,33221743136066636436,947415638925100675208,28436953641282225835143
%N Number of connected 3-regular (cubic) multigraphs on 2n unlabeled nodes rooted at an unoriented edge (or loop) whose removal does not disconnect the graph, loops allowed.
%C a(0) = 1 by convention. Loops add two to the degree of a node.
%C Instead of a rooted edge, the graph can be considered to have a pair of external legs (or half-edges). The external legs add 1 to the degree of a node, but do not contribute to the connectivity of the graph.
%C The 4-regular version of this sequence is A361135 since removing a single edge from a connected even degree regular graph cannot disconnect the graph.
%F G.f.: B(x) - x*(B(x)^2 + B(x^2))/2 where B(x) is the g.f. of A361412.
%e The illustrations in A352175 by _R. J. Mathar_ show 1, 2, 9, and 49 connected graphs corresponding to the initial terms of this sequence.
%Y Cf. A005967 (unrooted), A129427, A352175, A361135, A361412, A361446, A361448.
%K nonn
%O 0,2
%A _Andrew Howroyd_, Mar 12 2023
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