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A350528
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Triangle read by rows: T(n,k) is the number of labeled quasi-threshold graphs on vertex set [n] with k components, for n >= 1 and 1 <= k <= n.
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0
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1, 1, 1, 4, 3, 1, 23, 19, 6, 1, 181, 155, 55, 10, 1, 1812, 1591, 600, 125, 15, 1, 22037, 19705, 7756, 1750, 245, 21, 1, 315569, 286091, 116214, 27741, 4270, 434, 28, 1, 5201602, 4766823, 1983745, 493794, 81291, 9198, 714, 36, 1
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OFFSET
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1,4
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COMMENTS
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The family of quasi-threshold graphs is the smallest family of graphs that contains K_1 (a single vertex), and is closed under taking unions and adding dominating vertices (adjacent to all other vertices).
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LINKS
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FORMULA
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T(n,k) = Sum_{j=1..n} (-1)^(n-j)*Stirling2(n, j)*k*binomial(j, k)*j^(j-k-1) for n >= 1, 1 <= k <= n.
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EXAMPLE
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Triangle begins:
1;
1, 1;
4, 3, 1;
23, 19, 6, 1;
181, 155, 55, 10, 1;
1812, 1591, 600, 125, 15, 1;
22037, 19705, 7756, 1750, 245, 21, 1;
315569, 286091; 116214, 27741, 4270, 434, 28, 1;
...
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MATHEMATICA
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T[n_, k_] := T[n, k] = Sum[((-1)^(n - j))*StirlingS2[n, j]*k*Binomial[j, k]*(j^(j - k - 1)), {j, 1, n}]; Table[T[n, k], {n, 1, 12}, {k, 1, n}]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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