A361447 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.

1, 2, 9, 49, 338, 2744, 26025, 282419, 3463502, 47439030, 718618117, 11937743088, 215896959624, 4224096594516, 88919920910684, 2004237153640098, 48165411560792500, 1229462431057436457, 33221743136066636436, 947415638925100675208, 28436953641282225835143

Offset: 0

Author

Andrew Howroyd, Mar 12 2023

Keywords

• nonn

Comments

a(0) = 1 by convention. Loops add two to the degree of a node.

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.

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.

Examples

			The illustrations in A352175 by _R. J. Mathar_ show 1, 2, 9, and 49 connected graphs corresponding to the initial terms of this sequence.
		

Crossrefs

Cf. A005967 (unrooted), A129427, A352175, A361135, A361412, A361446, A361448.

Formula

G.f.: B(x) - x*(B(x)^2 + B(x^2))/2 where B(x) is the g.f. of A361412.