cp's OEIS Frontend

A119741 A008279, with the first and last of each row removed.

%I A119741 #26 Aug 22 2025 20:02:04
%S A119741 2,3,6,4,12,24,5,20,60,120,6,30,120,360,720,7,42,210,840,2520,5040,8,
%T A119741 56,336,1680,6720,20160,40320,9,72,504,3024,15120,60480,181440,362880,
%U A119741 10,90,720,5040,30240,151200,604800,1814400,3628800,11,110,990,7920,55440,332640,1663200,6652800,19958400,39916800
%N A119741 A008279, with the first and last of each row removed.
%C A119741 Triangle read by rows: T(n,k) (n>=2, k=1..n-1) is the number of topologies t on n points having exactly k+2 open sets such that t contains exactly one open set of size m for each m in {0,1,2,...,s,n} where s is the size of the largest proper open set in t. - _N. J. A. Sloane_, Jan 29 2016 [clarified by _Geoffrey Critzer_, Feb 19 2017]
%H A119741 Andrew Howroyd, <a href="/A119741/b119741.txt">Table of n, a(n) for n = 2..1276</a> (first 50 rows)
%H A119741 G. A. Kamel, <a href="http://www.aascit.org/journal/archive2?journalId=928&amp;paperId=2310">Partial Chain Topologies on Finite Sets</a>, Computational and Applied Mathematics Journal. Vol. 1, No. 4, 2015, pp. 174-179.
%F A119741 a(n) = (A003057(n))!/(A004736(n))! = (A002260(n))!*(A014410(n)).
%F A119741 T(n,k) = A173333(n+1,n-k+1), 1<=k<=n. - _Reinhard Zumkeller_, Feb 19 2010
%e A119741 Triangle begins:
%e A119741    2;
%e A119741    3,  6;
%e A119741    4, 12,  24;
%e A119741    5, 20,  60,  120;
%e A119741    6, 30, 120,  360,   720;
%e A119741    7, 42, 210,  840,  2520,   5040;
%e A119741    8, 56, 336, 1680,  6720,  20160,  40320;
%e A119741    9, 72, 504, 3024, 15120,  60480, 181440,  362880;
%e A119741   10, 90, 720, 5040, 30240, 151200, 604800, 1814400, 3628800;
%e A119741   ...
%p A119741 T:= (n, k)-> n!/(n-k)!:
%p A119741 seq(seq(T(n,k), k=1..n-1), n=2..11);  # _Alois P. Heinz_, Aug 22 2025
%t A119741 Table[FactorialPower[n, k], {n, 2, 11}, {k, 1, n-1}] // Flatten (* _Jean-François Alcover_, Feb 21 2020 *)
%Y A119741 Row sums give A038156.
%Y A119741 Triangles in this series: A268216, A268217, A268221, A268222, A268223.
%Y A119741 Cf. A008279, A014631, A282507.
%K A119741 nonn,tabl,changed
%O A119741 2,1
%A A119741 _Lekraj Beedassy_, Jul 29 2006
%E A119741 Edited by _Don Reble_, Aug 01 2006