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 A268217 Triangle read by rows: T(n,k) (n>=3, k=3..n+1) is the number of topologies t on n points having exactly k open sets such that t contains exactly one open set of size m for each m in {0,2,3,4,...,s,n} where s is the size of the largest proper open set in t. 6
 3, 6, 12, 10, 30, 60, 15, 60, 180, 360, 21, 105, 420, 1260, 2520, 28, 168, 840, 3360, 10080, 20160, 36, 252, 1512, 7560, 30240, 90720, 181440, 45, 360, 2880, 20160, 120960, 483840, 1451520, 2903040 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 3,1 COMMENTS When two leading 0's are added and last element repeated, rows give the coefficients of the path polynomials of the complete graph K_n. - Eric W. Weisstein, Jun 04 2017 LINKS G. A. Kamel, Partial Chain Topologies on Finite Sets, Computational and Applied Mathematics Journal. Vol. 1, No. 4, 2015, pp. 174-179. Eric Weisstein's World of Mathematics, Complete Graph Eric Weisstein's World of Mathematics, Graph Path EXAMPLE Triangle begins:    3;    6,  12;   10,  30,   60;   15,  60,  180,   360;   21, 105,  420,  1260,   2520;   28, 168,  840,  3360,  10080,  20160;   36, 252, 1512,  7560,  30240,  90720,  181440;   45, 360, 2880, 20160, 120960, 483840, 1451520, 2903040;   ... MATHEMATICA i = 2; Table[Table[Binomial[n, i] FactorialPower[n - i, k], {k, 0, n - i - 1}], {n, 2, 9}] // Grid (* Geoffrey Critzer, Feb 19 2017 *) CoefficientList[Table[-(1/2) (n - 1) n x^(n - 2) (Gamma[n - 1] - E^(1/x) Gamma[n - 1, 1/x]), {n, 3, 10}] // FunctionExpand, x] // Flatten (* Eric W. Weisstein, Jun 04 2017 *) CROSSREFS Row sums give A038158. Triangles in this series: A268216, A268217, A268221, A268222, A268223. Cf. A282507. Sequence in context: A342786 A293474 A308727 * A254793 A182633 A038587 Adjacent sequences:  A268214 A268215 A268216 * A268218 A268219 A268220 KEYWORD nonn,tabl,more AUTHOR N. J. A. Sloane, Jan 29 2016 EXTENSIONS Title clarified by Geoffrey Critzer, Feb 19 2017 STATUS approved

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Last modified September 25 13:37 EDT 2021. Contains 347654 sequences. (Running on oeis4.)