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A350746
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Triangle read by rows: T(n,k) is the number of labeled quasi-loop-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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2, 3, 4, 16, 18, 8, 133, 155, 72, 16, 1521, 1810, 910, 240, 32, 22184, 26797, 14145, 4180, 720, 64, 393681, 480879, 262514, 83230, 16520, 2016, 128, 8233803, 10144283, 5675866, 1888873, 409360, 58912, 5376, 256
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OFFSET
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1,1
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COMMENTS
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The family of quasi-loop-threshold graphs is the smallest family of looped graphs that contains K_1 (a single vertex) and K^loop_1 (a single looped vertex), and is closed under taking unions and adding looped dominating vertices (looped, and adjacent to everything previously added).
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LINKS
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FORMULA
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See Section 1.4 of Galvin, Wesley and Zacovic link for two methods to compute T(n,k).
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EXAMPLE
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Triangle begins:
2;
3, 4;
16, 18, 8;
133, 155, 72, 16;
1521, 1810, 910, 240, 32;
22184, 26797, 14145, 4180, 720, 64;
393681, 480879, 262514, 83230, 16520, 2016, 128;
8233803, 10144283, 5675866, 1888873, 409360, 58912, 5376, 256;
...
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MATHEMATICA
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qltconn[0] = 0; qltconn[1] = 2; qltconn[n_] := qltconn[n] = Sum[StirlingS2[n, k]*(k^(k - 1)), {k, 1, n}] (*qltconn is the number of connected quasi loop threshold graphs on n vertices*); T[n_, l_] := T[n, l] := (Factorial[n]/Factorial[l])*Coefficient[(Sum[(qltconn[k]*(x^k))/Factorial[k], {k, 1, n}])^l, x, n]; Table[T[n, l], {n, 1, 12}, {l, 1, n}]
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CROSSREFS
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Except at n=1, the first column is A048802 (A048802 takes value 1 at n=1).
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KEYWORD
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AUTHOR
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STATUS
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approved
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