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A369195
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Irregular triangle read by rows where T(n,k) is the number of labeled connected loop-graphs covering n vertices with k edges.
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4
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1, 0, 1, 0, 1, 2, 1, 0, 0, 3, 10, 12, 6, 1, 0, 0, 0, 16, 79, 162, 179, 116, 45, 10, 1, 0, 0, 0, 0, 125, 847, 2565, 4615, 5540, 4720, 2948, 1360, 455, 105, 15, 1, 0, 0, 0, 0, 0, 1296, 11436, 47100, 121185, 220075, 301818, 325578, 282835, 200115, 115560, 54168, 20343, 5985, 1330, 210, 21, 1
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OFFSET
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0,6
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COMMENTS
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This sequence excludes the graph consisting of a single isolated vertex without a loop. - Andrew Howroyd, Feb 02 2024
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LINKS
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FORMULA
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E.g.f.: 1 - x + log(Sum_{j >= 0} (1 + y)^binomial(j+1, 2)*x^j/j!). - Andrew Howroyd, Feb 02 2024
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EXAMPLE
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Triangle begins:
1
0 1
0 1 2 1
0 0 3 10 12 6 1
0 0 0 16 79 162 179 116 45 10 1
Row n = 3 counts the following loop-graphs (loops shown as singletons):
. . {12,13} {1,12,13} {1,2,12,13} {1,2,3,12,13} {1,2,3,12,13,23}
{12,23} {1,12,23} {1,2,12,23} {1,2,3,12,23}
{13,23} {1,13,23} {1,2,13,23} {1,2,3,13,23}
{2,12,13} {1,3,12,13} {1,2,12,13,23}
{2,12,23} {1,3,12,23} {1,3,12,13,23}
{2,13,23} {1,3,13,23} {2,3,12,13,23}
{3,12,13} {1,12,13,23}
{3,12,23} {2,3,12,13}
{3,13,23} {2,3,12,23}
{12,13,23} {2,3,13,23}
{2,12,13,23}
{3,12,13,23}
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MATHEMATICA
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csm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}], Length[Intersection@@s[[#]]]>0&]}, If[c=={}, s, csm[Sort[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n], {1, 2}], {k}], Length[Union@@#]==n&&Length[csm[#]]<=1&]], {n, 0, 5}, {k, 0, Binomial[n+1, 2]}]
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PROG
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(PARI) T(n)={[Vecrev(p) | p<-Vec(serlaplace(1 - x + log(sum(j=0, n, (1 + y)^binomial(j+1, 2)*x^j/j!, O(x*x^n))))) ]}
{ my(A=T(6)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Feb 02 2024
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CROSSREFS
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A000666 counts unlabeled loop-graphs.
A006125 counts simple graphs, also loop-graphs if shifted left.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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