A321305 Triangle T(n,f): the number of signed cubic graphs on 2n vertices with f edges of the first sign.

1, 0, 0, 0, 0, 1, 1, 2, 3, 2, 1, 1, 2, 3, 8, 16, 21, 21, 16, 8, 3, 2, 5, 14, 57, 152, 313, 474, 551, 474, 313, 152, 57, 14, 5, 19, 91, 491, 1806, 5034, 10604, 17318, 22033, 22033, 17318, 10604, 5034, 1806, 491, 91, 19, 85, 706, 4981, 23791, 84575, 229078, 487020, 825127, 1127783, 1250632, 1127783, 825127, 487020, 229078, 84575, 23791, 4981, 706, 85

Offset: 0

Author

R. J. Mathar, Nov 03 2018

Keywords

• nonn
• tabf

Comments

These are connected, undirected, simple cubic graphs where each edge is signed as either "+" or "-". Row n has 1+3n entries, 0<=f<=3n. The column f=0 (1, 0, 1, 2, 5,...) counts the cubic graphs (A002851). The column f=1 (0, 1, 3, 14, 91, 706,...) counts the edge-rooted cubic graphs.

Examples

			The triangle starts:
0 vertices: 1
2 vertices: 0,0,0,0
4 vertices: 1,1,2,3,2,1,1
6 vertices: 2,3,8,16,21,21,16,8,3,2
8 vertices: 5,14,57,152,313,474,551,474,313,152,57,14,5
10 vertices: 19,91,491,1806,5034,10604,17318,22033,22033,17318,10604,5034,1806,491,91,19
		

Crossrefs

Cf. A002851 (first column), A321304 (signed vertices), A302939 (signed trees).

Formula

T(n,f) = T(n,3*n-f).