A321306 The number of connected weighted cubic graphs with weight n on 6 vertices.

2, 2, 7, 12, 26, 41, 76, 113, 183, 264, 393, 543, 768, 1024, 1385, 1801, 2355, 2989, 3811, 4740, 5911, 7234, 8857, 10680, 12883, 15336, 18254, 21496, 25293, 29491, 34361, 39713, 45860, 52598, 60260, 68627, 78079, 88354, 99882, 112385, 126316, 141379, 158082, 176080

Offset: 6

Author

R. J. Mathar, Nov 03 2018

Keywords

• nonn
• easy

Comments

Each vertex of the 2 simple cubic graphs is assigned an integer number (weight) >=1. The weight of the graph is the sum of the weights of the vertices.

The cycle indices of the permutation group of vertex permutations of the two cubic graphs on 6 vertices are ( +t[1]^6 +3*t[1]^2*t[2]^2 +2*t[3]^2 +4*t[2]^3 +2*t[6])/12 and +( +t[1]^6 +6*t[1]^4*t[2] +9*t[1]^2*t[2]^2 +4*t[1]^3*t[3] +12*t[1]*t[2]*t[3] +6*t[2]^3 +18*t[2]*t[4] +12*t[6] +4*t[3]^2)/72 . The ordinary generating function of the sequence is obtained by adding the two cycle indices and setting t[i] -> x^i/(1-x^i).

Examples

			a(6)=2 because there are 2 cubic graphs (see A002851), and if the weight is the same as the number of vertices, there is one case for each.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 6..1000
• Index entries for linear recurrences with constant coefficients, signature (2,0,-1,0,-2,3,-2,1,1,-2,3,-2,0,-1,0,2,-1).

Crossrefs

Cf. A026810 (4 vertices), A321307 (8 vertices), A005513.

Formula

G.f.: (x^10 +3*x^8 -x^7 +4*x^6 +4*x^4 +3*x^2 -2*x+2) *x^6/((-1+x)^6 *(1+x)^3 *(1+x^2) *(x^2+x+1)^2 *(x^2-x+1)).

Extensions

Terms a(36) and beyond from Andrew Howroyd, Apr 27 2020