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A268217
Triangle read by rows: T(n,k) (n>=3, k=3..n) 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, 2520, 15120, 75600, 302400, 907200, 1814400, 55, 495, 3960, 27720, 166320, 831600, 3326400, 9979200, 19958400
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
Andrew Howroyd, Table of n, a(n) for n = 3..1277 (first 50 rows)
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, 2520, 15120, 75600, 302400, 907200, 1814400;
...
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: A119741, A268217, A268221, A268222, A268223.
Cf. A282507.
Sequence in context: A342786 A293474 A308727 * A254793 A353715 A182633
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Jan 29 2016
EXTENSIONS
Title clarified by Geoffrey Critzer, Feb 19 2017
Corrected and extended by Andrew Howroyd, Aug 09 2025
STATUS
approved