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 A350528 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. 0
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS 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). LINKS D. Galvin, G. Wesley and B. Zacovic, Enumerating threshold graphs and some related graph classes, arXiv:2110.08953 [math.CO], 2021. FORMULA 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. EXAMPLE 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; ... MATHEMATICA 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}] CROSSREFS First column is A058863. Row sums are A058864. Cf. A008277. Sequence in context: A128320 A189507 A348436 * A208057 A298673 A245732 Adjacent sequences: A350525 A350526 A350527 * A350529 A350530 A350531 KEYWORD nonn,tabl AUTHOR David Galvin, Jan 03 2022 STATUS approved

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Last modified January 29 00:02 EST 2023. Contains 359905 sequences. (Running on oeis4.)