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A281779 Number of distinct topologies on an n-set that have exactly 11 open sets.

%I A281779 #23 Aug 01 2024 14:12:02
%S A281779 0,0,0,0,0,500,16980,342160,5486040,77926380,1031160060,13047426920,
%T A281779 160124426880,1921105846660,22632779709540,262513678889280,
%U A281779 3002768326532520,33914184260797340,378596540805849420,4181330954328313240,45727913513193402960,495618273676457274420
%N A281779 Number of distinct topologies on an n-set that have exactly 11 open sets.
%H A281779 Colin Barker, <a href="/A281779/b281779.txt">Table of n, a(n) for n = 0..950</a>
%H A281779 Moussa Benoumhani, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Benoumhani/benoumhani11.html">The Number of Topologies on a Finite Set</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.6.
%H A281779 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (55,-1320,18150,-157773,902055,-3416930,8409500,-12753576,10628640,-3628800).
%F A281779 a(n) = 25/6*5! Stirling2(n, 5) + 79/6*6! Stirling2(n, 6) + 29/2*7! Stirling2(n, 7) + 39/4*8! Stirling2(n, 8) + 4*9! Stirling2(n, 9) + 10! Stirling2(n, 10).
%F A281779 G.f.: 20*x^5*(25 - 526*x + 3413*x^2 + 292*x^3 - 72756*x^4 + 226800*x^5) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)*(1 - 8*x)*(1 - 9*x)*(1 - 10*x)). - _Colin Barker_, Jan 30 2017
%o A281779 (PARI) a(n) = 25*5!*stirling(n, 5, 2)/6 + 79*6!*stirling(n, 6, 2)/6 + 29*7!*stirling(n, 7, 2)/2 + 39*8!*stirling(n, 8, 2)/4 + 4*9!*stirling(n, 9, 2) + 10!*stirling(n, 10, 2) \\ _Colin Barker_, Jan 30 2017
%o A281779 (PARI) concat(vector(4), Vec(20*x^5*(25 - 526*x + 3413*x^2 + 292*x^3 - 72756*x^4 + 226800*x^5) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)*(1 - 8*x)*(1 - 9*x)*(1 - 10*x)) + O(x^30))) \\ _Colin Barker_, Jan 30 2017
%Y A281779 The number of distinct topologies on an n-set with exactly k open sets for k=2..12 is given by A000012, A000918, A281773, A028244, A281774, A281775, A281776, A281777,A281778, A281779, A281780.
%K A281779 nonn,easy
%O A281779 0,6
%A A281779 _Geoffrey Critzer_, Jan 29 2017