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%I #21 Aug 03 2021 15:31:53
%S 0,0,0,1,54,955,11760,122941,1175034,10595215,91506420,763624081,
%T 6194818014,49084747075,381338401080,2914184784421,21965095364994,
%U 163656285828535,1207613518375740,8838842878371961,64253768864671974,464416229729871595,3340518964319750400
%N Number of distinct topologies on an n-set that have exactly 8 open sets.
%H Colin Barker, <a href="/A281776/b281776.txt">Table of n, a(n) for n = 0..1000</a>
%H 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 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (28,-322,1960,-6769,13132,-13068,5040).
%F a(n) = Stirling2(n, 3) + 2*4! Stirling2(n, 4) + 15/4*5! Stirling2(n, 5) + 5/2*6! Stirling2(n, 6) + 7! Stirling2(n, 7).
%F From _Colin Barker_, Jan 30 2017: (Start)
%F a(n) = 13/4 - 19*2^(n-1) + 44*3^(n-1) - 2^(n-1)*3^(2+n) - 57*4^(n-1) + (39*5^n)/4 + 7^n for n>0.
%F G.f.: x^3*(1 + 26*x - 235*x^2 + 448*x^3 + 2100*x^4) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)).
%F (End)
%o (PARI) concat(vector(3), Vec(x^3*(1 + 26*x - 235*x^2 + 448*x^3 + 2100*x^4) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)) + O(x^30))) \\ _Colin Barker_, Jan 30 2017
%Y 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 nonn,easy
%O 0,5
%A _Geoffrey Critzer_, Jan 29 2017
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