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%I #20 Jan 08 2018 01:58:32
%S 4,10,20,20,60,120,35,140,420,840,56,280,1120,3360,6720,84,504,2520,
%T 10080,30240,60480,120,840,5040,25200,100800,302400,604800,165,1320,
%U 9240,55440,277200,1108800,3326400,6652800,220,1980,15840,110880,665280,3326400,13305600,39916800,79833600
%N Triangle read by rows: T(n,k) (n>=4, 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 {0,3,4,5,...,s,n} where s is the size of the largest proper open set in t.
%H G. A. Kamel, <a href="http://www.aascit.org/journal/archive2?journalId=928&paperId=2310">Partial Chain Topologies on Finite Sets</a>, Computational and Applied Mathematics Journal. Vol. 1, No. 4, 2015, pp. 174-179.
%e Triangle begins:
%e 4,
%e 10,20,
%e 20,60,120,
%e 35,140,420,840,
%e 56,280,1120,3360,6720,
%e 84,504,2520,10080,30240,60480,
%e 120,840,5040,25200,100800,302400,604800,
%e ...
%t i = 3; Table[Table[Binomial[n, i] FactorialPower[n - i, k], {k, 0, n - i - 1}], {n, 2, 12}] // Grid (* _Geoffrey Critzer_, Feb 19 2017 *)
%Y Row sums give A268218.
%Y Triangles in this series: A268216, A268217, A268221, A268222, A268223.
%Y Cf. A282507.
%K nonn,tabl,more
%O 4,1
%A _N. J. A. Sloane_, Jan 30 2016
%E Title clarified and more terms added by _Geoffrey Critzer_, Feb 19 2017
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