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A369199
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Irregular triangle read by rows where T(n,k) is the number of labeled loop-graphs covering n vertices with k edges.
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33
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1, 0, 1, 0, 1, 3, 1, 0, 0, 6, 17, 15, 6, 1, 0, 0, 3, 46, 150, 228, 206, 120, 45, 10, 1, 0, 0, 0, 45, 465, 1803, 3965, 5835, 6210, 4955, 2998, 1365, 455, 105, 15, 1, 0, 0, 0, 15, 645, 5991, 27364, 79470, 165555, 264050, 334713, 344526, 291200, 202860, 116190, 54258, 20349, 5985, 1330, 210, 21, 1
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OFFSET
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0,6
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LINKS
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FORMULA
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E.g.f.: exp(-x) * (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 3 1
0 0 6 17 15 6 1
0 0 3 46 150 228 206 120 45 10 1
Row n = 3 counts the following loop-graphs (loops shown as singletons):
{1,23} {1,2,3} {1,2,3,12} {1,2,3,12,13} {1,2,3,12,13,23}
{2,13} {1,2,13} {1,2,3,13} {1,2,3,12,23}
{3,12} {1,2,23} {1,2,3,23} {1,2,3,13,23}
{12,13} {1,3,12} {1,2,12,13} {1,2,12,13,23}
{12,23} {1,3,23} {1,2,12,23} {1,3,12,13,23}
{13,23} {1,12,13} {1,2,13,23} {2,3,12,13,23}
{1,12,23} {1,3,12,13}
{1,13,23} {1,3,12,23}
{2,3,12} {1,3,13,23}
{2,3,13} {1,12,13,23}
{2,12,13} {2,3,12,13}
{2,12,23} {2,3,12,23}
{2,13,23} {2,3,13,23}
{3,12,13} {2,12,13,23}
{3,12,23} {3,12,13,23}
{3,13,23}
{12,13,23}
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MATHEMATICA
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Table[Length[Select[Subsets[Subsets[Range[n], {1, 2}], {k}], Length[Union@@#]==n&]], {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(exp(-x + O(x*x^n))*(sum(j=0, n, (1 + y)^binomial(j+1, 2)*x^j/j!)))) ]}
{ 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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The version without loops is A054548.
This is the covering case of A084546.
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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