A321305 - OEIS

%I #10 Nov 05 2018 06:11:47

%S 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,

%T 551,474,313,152,57,14,5,19,91,491,1806,5034,10604,17318,22033,22033,

%U 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

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

%C 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.

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

%e The triangle starts:

%e 0 vertices: 1

%e 2 vertices: 0,0,0,0

%e 4 vertices: 1,1,2,3,2,1,1

%e 6 vertices: 2,3,8,16,21,21,16,8,3,2

%e 8 vertices: 5,14,57,152,313,474,551,474,313,152,57,14,5

%e 10 vertices: 19,91,491,1806,5034,10604,17318,22033,22033,17318,10604,5034,1806,491,91,19

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

%K nonn,tabf

%O 0,8

%A _R. J. Mathar_, Nov 03 2018