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 A350745 Triangle read by rows: T(n,k) is the number of labeled loop-threshold graphs on vertex set [n] with k loops, for n >= 0 and 0 <= k <= n. 1
 1, 1, 1, 1, 4, 1, 1, 12, 12, 1, 1, 32, 84, 32, 1, 1, 80, 460, 460, 80, 1, 1, 192, 2190, 4600, 2190, 192, 1, 1, 448, 9534, 37310, 37310, 9534, 448, 1, 1, 1024, 39032, 264208, 483140, 264208, 39032, 1024, 1, 1, 2304, 152856, 1702344, 5229756, 5229756, 1702344, 152856, 2304, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS 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). LINKS Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50). D. Galvin, G. Wesley and B. Zacovic, Enumerating threshold graphs and some related graph classes, arXiv:2110.08953 [math.CO], 2021. FORMULA T(n,0) = 1; T(n,k) = binomial(n,k) * Sum_{j=1..n} j!*Stirling2(k,j) * ((j-1)! * Stirling2(n-k,j-1) + 2*j!*Stirling2(n-k,j) + (j+1)!*Stirling2(n-k,j+1)). T(n,k) = T(n,n-k). Sum_{k=0..2*n} (-1)^k * T(2*n,k) = A210657(n). - Alois P. Heinz, Feb 01 2022 EXAMPLE Triangle begins: 1; 1, 1; 1, 4, 1; 1, 12, 12, 1; 1, 32, 84, 32, 1; 1, 80, 460, 460, 80, 1; 1, 192, 2190, 4600, 2190, 192, 1; 1, 448, 9534, 37310, 37310, 9534, 448, 1; 1, 1024, 39032, 264208, 483140, 264208, 39032, 1024, 1; 1, 2304, 152856, 1702344, 5229756, 5229756, 1702344, 152856, 2304, 1; ... MATHEMATICA T[n_, 0] := T[n, 0] = 1; T[n_, k_] := T[n, k] = Binomial[n, k]*Sum[Factorial[l]*StirlingS2[k, l]*(Factorial[l - 1]*StirlingS2[n - k, l - 1] + 2*Factorial[l]*StirlingS2[n - k, l] + Factorial[l + 1]*StirlingS2[n - k, l + 1]), {l, 1, n + 1}]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] PROG (PARI) T(n, k) = if(k==0, 1, binomial(n, k) * sum(j=1, n, j!*stirling(k, j, 2) * ((j-1)! * stirling(n-k, j-1, 2) + 2*j!*stirling(n-k, j, 2) + (j+1)!*stirling(n-k, j+1, 2)))) \\ Andrew Howroyd, May 06 2023 CROSSREFS Row sums are A000629. Columns k=0..1 give: A000012, A001787, Cf. A210657. Sequence in context: A099759 A350819 A072590 * A111636 A220688 A146990 Adjacent sequences: A350742 A350743 A350744 * A350746 A350747 A350748 KEYWORD nonn,tabl AUTHOR David Galvin, Jan 13 2022 STATUS approved

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Last modified June 12 23:24 EDT 2024. Contains 373362 sequences. (Running on oeis4.)