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

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, 542, 931, 1052, 931, 542, 248, 64, 19, 85, 490, 2382, 7011, 15199, 23405, 27336, 23405, 15199, 7011, 2382, 490, 85, 509, 4595, 27233, 101002, 268675, 523246, 776657, 882321, 776657, 523246, 268675, 101002, 27233, 4595, 509

Offset: 0

Author

R. J. Mathar, Nov 03 2018

Keywords

• nonn
• tabf

Comments

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.

Examples

			The triangle starts:
0 vertices:   1;
2 vertices:   0,  0,   0;
4 vertices:   1,  1,   1,   1,   1;
6 vertices:   2,  2,   5,   5,   5,    2,   2;
8 vertices:   5, 10,  31,  46,  63,   46,  31,  10,   5;
10 vertices: 19, 64, 248, 542, 931, 1052, 931, 542, 248, 64, 19;
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..440 (rows 0..20)

Crossrefs

Columns f=0, 1, 2 are A002851, A361407, A361408.

Row sums are A361403.

Central coefficients are A361406.

Cf. A294783 (bicolored trees), A321305 (signed edges), A361361 (not necessarily connected).

Formula

T(n,f) = T(n,2n-f).

Extensions

Terms a(49) and beyond from Andrew Howroyd, Mar 11 2023