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A281777 Number of distinct topologies on an n-set that have exactly 9 open sets.

%I A281777 #21 Aug 03 2021 15:49:30
%S A281777 0,0,0,0,20,800,14260,189280,2181060,23241120,235737620,2308206560,
%T A281777 21979728100,204477713440,1864504348980,16707856095840,
%U A281777 147469451067140,1284607771225760,11063319237792340,94343562846289120,797685042851814180,6694943490279586080
%N A281777 Number of distinct topologies on an n-set that have exactly 9 open sets.
%H A281777 Colin Barker, <a href="/A281777/b281777.txt">Table of n, a(n) for n = 0..1000</a>
%H A281777 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 A281777 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (36,-546,4536,-22449,67284,-118124,109584,-40320).
%F A281777 a(n) = 5/6*4! Stirling2(n, 4) + 5*5! Stirling2(n, 5) + 11/2*6! Stirling2(n, 6) + 3*7! Stirling2(n, 7) + 8! Stirling2(n, 8).
%F A281777 G.f.: 20*x^4*(1 + 4*x - 181*x^2 + 1100*x^3 - 1344*x^4) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)*(1 - 8*x)). - _Colin Barker_, Jan 30 2017
%t A281777 LinearRecurrence[{36,-546,4536,-22449,67284,-118124,109584,-40320},{0,0,0,0,20,800,14260,189280,2181060},30] (* _Harvey P. Dale_, Aug 19 2020 *)
%o A281777 (PARI) concat(vector(4), Vec(20*x^4*(1 + 4*x - 181*x^2 + 1100*x^3 - 1344*x^4) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)*(1 - 8*x)) + O(x^30))) \\ _Colin Barker_, Jan 30 2017
%Y A281777 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 A281777 nonn,easy
%O A281777 0,5
%A A281777 Submitted on behalf of Moussa Benoumhani by _Geoffrey Critzer_, Jan 29 2017