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A281775 Number of distinct topologies on an n-set that have exactly 7 open sets.

%I A281775 #27 Aug 03 2021 15:10:44
%S A281775 0,0,0,0,54,780,7830,67620,535374,3992940,28483110,196316340,
%T A281775 1317106494,8650141500,55853351190,355770438660,2241509994414,
%U A281775 13998294536460,86795899256070,535048203626580,3282628800655134,20061393719417820,122212221633141750
%N A281775 Number of distinct topologies on an n-set that have exactly 7 open sets.
%H A281775 Colin Barker, <a href="/A281775/b281775.txt">Table of n, a(n) for n = 0..1000</a>
%H A281775 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 A281775 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (21,-175,735,-1624,1764,-720).
%F A281775 a(n) = 9/4*4! Stirling2(n, 4) + 2*5! Stirling2(n, 5) + 6! Stirling2(n, 6).
%F A281775 From _Colin Barker_, Jan 30 2017: (Start)
%F A281775 G.f.: 6*x^4*(9 - 59*x + 150*x^2)/((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)).
%F A281775 a(n) = 21*a(n-1) - 175*a(n-2) + 735*a(n-3) - 1624*a(n-4) + 1764*a(n-5) - 720*a(n-6) for n>6.
%F A281775 a(n) = -5 + 17*2^(n-1) - 3^(2+n) + 29*4^(n-1) - 4*5^n + 6^n for n>0. (End)
%o A281775 (PARI) a(n) = 9*4!*stirling(n, 4, 2)/4 + 2*5!*stirling(n, 5, 2) + 6!*stirling(n, 6, 2) \\ _Colin Barker_, Jan 30 2017
%o A281775 (PARI) concat(vector(4), Vec(6*x^4*(9 - 59*x + 150*x^2) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)) + O(x^30))) \\ _Colin Barker_, Jan 30 2017
%Y A281775 The number of distinct topologies on an n-set with exactly k open sets for k=2..12 is given by A000012, A000918, A281773, A281774, A028244, A281775, A281776, A281777, A281778, A281779, A281780.
%K A281775 nonn,easy
%O A281775 0,5
%A A281775 Submitted on behalf of Moussa Benoumhani by _Geoffrey Critzer_, Jan 29 2017