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 A348436 Triangle read by rows. T(n,k) is the number of labeled threshold graphs on n vertices with k components, for 1 <= k <= n. 0
 1, 1, 1, 4, 3, 1, 23, 16, 6, 1, 166, 115, 40, 10, 1, 1437, 996, 345, 80, 15, 1, 14512, 10059, 3486, 805, 140, 21, 1, 167491, 116096, 40236, 9296, 1610, 224, 28, 1, 2174746, 1507419, 522432, 120708, 20916, 2898, 336, 36, 1, 31374953, 21747460, 7537095, 1741440, 301770, 41832, 4830, 480, 45, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS The class of threshold graphs is the smallest class of graphs that includes K1 and is closed under adding isolated vertices and dominating vertices. LINKS Table of n, a(n) for n=1..55. D. Galvin, G. Wesley and B. Zacovic, Enumerating threshold graphs and some related graph classes, arXiv:2110.08953 [math.CO], 2021. Sam Spiro, Counting Threshold Graphs with Eulerian Numbers, arXiv:1909.06518 [math.CO], 2019. FORMULA T(1,1) = 1; for n >= 2, T(n,1) = A005840(n)/2; for n >= 3 and 2 <= k <= n-1, T(n,k) = binomial(n,k-1)*T(n-k+1,1); and for n >= 2, T(n,n)=1. T(n, k) = binomial(n, k-1)*A053525(n - k + 1) if k != n, otherwise 1. - Peter Luschny, Oct 24 2021 EXAMPLE Triangle begins: 1; 1, 1; 4, 3, 1; 23, 16, 6, 1; 166, 115, 40, 10, 1; 1437, 996, 345, 80, 15, 1; 14512, 10059, 3486, 805, 140, 21, 1; 167491, 116096, 40236, 9296, 1610, 224, 28, 1; 2174746, 1507419, 522432, 120708, 20916, 2898, 336, 36, 1; 31374953, 21747460, 7537095, 1741440, 301770, 41832, 4830, 480, 45, 1; ... MAPLE T := (n, k) -> `if`(n = k, 1, binomial(n, k-1)*A053525(n-k+1)): for n from 1 to 10 do seq(T(n, k), k=1..n) od; # Peter Luschny, Oct 24 2021 MATHEMATICA eulerian[0, 0] := 1; eulerian[n_, m_] := eulerian[n, m] = Sum[((-1)^k)*Binomial[n + 1, k]*((m + 1 - k)^n), {k, 0, m + 1}]; (* t[n] counts the labeled threshold graphs on n vertices *) t[0] = 1; t[1] = 1; t[n_] := t[n] = Sum[(n - k)*eulerian[n - 1, k - 1]*(2^k), {k, 1, n - 1}]; T[1, 1] := 1; T[n_, 1] := T[n, 1] = (1/2)*t[n]; T[n_, n_] := T[n, n] = 1; T[n_, k_] := T[n, k] = Binomial[n, k - 1]*T[n - k + 1, 1]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten CROSSREFS Cf. A005840 (row sums), A317057 (column k=1), A053525. Sequence in context: A181355 A128320 A189507 * A350528 A208057 A298673 Adjacent sequences: A348433 A348434 A348435 * A348437 A348438 A348439 KEYWORD nonn,tabl AUTHOR David Galvin, Oct 18 2021 STATUS approved

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Last modified December 3 19:10 EST 2023. Contains 367540 sequences. (Running on oeis4.)