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A321306
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The number of connected weighted cubic graphs with weight n on 6 vertices.
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3
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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
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OFFSET
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6,1
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COMMENTS
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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).
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LINKS
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FORMULA
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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)).
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EXAMPLE
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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.
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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