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

%I A321305 #10 Nov 05 2018 06:11:47
%S A321305 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 A321305 551,474,313,152,57,14,5,19,91,491,1806,5034,10604,17318,22033,22033,
%U A321305 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 A321305 Triangle T(n,f): the number of signed cubic graphs on 2n vertices with f edges of the first sign.
%C A321305 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 A321305 T(n,f) = T(n,3*n-f).
%e A321305 The triangle starts:
%e A321305 0 vertices: 1
%e A321305 2 vertices: 0,0,0,0
%e A321305 4 vertices: 1,1,2,3,2,1,1
%e A321305 6 vertices: 2,3,8,16,21,21,16,8,3,2
%e A321305 8 vertices: 5,14,57,152,313,474,551,474,313,152,57,14,5
%e A321305 10 vertices: 19,91,491,1806,5034,10604,17318,22033,22033,17318,10604,5034,1806,491,91,19
%Y A321305 Cf. A002851 (first column), A321304 (signed vertices), A302939 (signed trees).
%K A321305 nonn,tabf
%O A321305 0,8
%A A321305 _R. J. Mathar_, Nov 03 2018