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A350531
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Triangle read by rows: T(n,k) is the number of labeled loop-threshold graphs on vertex set [n] with k components, for n >= 1 and 1 <= k <= n.
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1
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2, 3, 3, 13, 9, 4, 75, 52, 18, 5, 541, 375, 130, 30, 6, 4683, 3246, 1125, 260, 45, 7, 47293, 32781, 11361, 2625, 455, 63, 8, 545835, 378344, 131124, 30296, 5250, 728, 84, 9, 7087261, 4912515, 1702548, 393372, 68166, 9450, 1092, 108, 10
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OFFSET
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1,1
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COMMENTS
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Loop-threshold graphs are constructed from either a single unlooped vertex or a single looped vertex by iteratively adding isolated vertices (adjacent to nothing previously added) and looped dominating vertices (looped, and adjacent to everything previously added).
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LINKS
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FORMULA
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T(n,n) = n+1 for n >= 1; T(n,1) = Sum_{j=0..n-1} A173018(n,j)*2^j for n >= 2; T(n,k) = binomial(n, k-1)*T(n-k+1,1) for n >= 3, 2 <= k <= n-1.
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EXAMPLE
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Triangle begins:
2;
3, 3;
13, 9, 4;
75, 52, 18, 5;
541, 375, 130, 30, 6;
4683, 3246, 1125, 260, 45, 7;
47293, 32781, 11361, 2625, 455, 63, 8;
545835, 378344, 131124, 30296, 5250, 728, 84, 9;
7087261, 4912515, 1702548, 393372, 68166, 9450, 1092, 108, 10;
...
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MATHEMATICA
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eulerian[n_, m_] := eulerian[n, m] =
Sum[((-1)^k)*Binomial[n+1, k]*((m+1-k)^n), {k, 0, m+1}] (* eulerian[n, m] is an Eulerian number, counting the permutations of [n] with m descents *); T[1, 1] = 2; T[n_, 1] := T[n, 1] = Sum[eulerian[n, k]*(2^k), {k, 0, n - 1}]; T[n_, n_] := T[n, n] = n + 1; T[n_, k_] := T[n, k] = Binomial[n, k - 1]*T[n - k + 1, 1]; Table[T[n, k], {n, 1, 12}, {k, 1, n}]
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CROSSREFS
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Except at n = 1, first column is A000670.
Essentially the same as A154921 --- in A350531 (this triangle), replace the last nonzero entry in row m (this entry is m+1) with the two entries m, 1 to get A154921.
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KEYWORD
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AUTHOR
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STATUS
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approved
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