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A321304 Triangle T(n,f): the number of bicolored connected cubic graphs on 2n vertices with f vertices of the first color.

%I A321304 #17 Sep 23 2023 07:31:10
%S A321304 1,0,0,0,1,1,1,1,1,2,2,5,5,5,2,2,5,10,31,46,63,46,31,10,5,19,64,248,
%T A321304 542,931,1052,931,542,248,64,19,85,490,2382,7011,15199,23405,27336,
%U A321304 23405,15199,7011,2382,490,85,509,4595,27233,101002,268675,523246,776657,882321,776657,523246,268675,101002,27233,4595,509
%N A321304 Triangle T(n,f): the number of bicolored connected cubic graphs on 2n vertices with f vertices of the first color.
%C A321304 These are connected, undirected, simple cubic graphs where each vertex has either the first or the second color. Row n has 2n+1 entries, 0<=f<=2n. The column f=0 (1, 0, 2, 5,...) counts the cubic graphs (A002851). The column f=1 (0, 1, 2, 10, 64, 490...) counts the rooted cubic graphs.
%H A321304 Andrew Howroyd, <a href="/A321304/b321304.txt">Table of n, a(n) for n = 0..440</a> (rows 0..20)
%F A321304 T(n,f) = T(n,2n-f).
%e A321304 The triangle starts:
%e A321304 0 vertices:   1;
%e A321304 2 vertices:   0,  0,   0;
%e A321304 4 vertices:   1,  1,   1,   1,   1;
%e A321304 6 vertices:   2,  2,   5,   5,   5,    2,   2;
%e A321304 8 vertices:   5, 10,  31,  46,  63,   46,  31,  10,   5;
%e A321304 10 vertices: 19, 64, 248, 542, 931, 1052, 931, 542, 248, 64, 19;
%Y A321304 Columns f=0, 1, 2 are A002851, A361407, A361408.
%Y A321304 Row sums are A361403.
%Y A321304 Central coefficients are A361406.
%Y A321304 Cf. A294783 (bicolored trees), A321305 (signed edges), A361361 (not necessarily connected).
%K A321304 nonn,tabf
%O A321304 0,10
%A A321304 _R. J. Mathar_, Nov 03 2018
%E A321304 Terms a(49) and beyond from _Andrew Howroyd_, Mar 11 2023