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A281774 Number of distinct topologies on an n-set with exactly 6 open sets.

%I A281774 #42 Aug 03 2021 15:11:04
%S A281774 0,0,0,6,72,630,4680,31206,193032,1131990,6386760,35025606,188061192,
%T A281774 993760950,5187840840,26831095206,137770476552,703455087510,
%U A281774 3576115150920,18117222864006,91536570671112,461496288791670,2322770028381000,11675109032796006
%N A281774 Number of distinct topologies on an n-set with exactly 6 open sets.
%H A281774 Colin Barker, <a href="/A281774/b281774.txt">Table of n, a(n) for n = 0..1000</a>
%H A281774 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 A281774 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (15,-85,225,-274,120).
%F A281774 a(n) = 3! Stirling2(n, 3) + 3/2*4! Stirling2(n, 4) + 5! Stirling2(n, 5).
%F A281774 From _Colin Barker_, Jan 30 2017: (Start)
%F A281774 a(n) = 2 - 2^(2+n) - 7*2^(2*n-1) + 5*3^n + 5^n for n>5.
%F A281774 a(n) = 15*a(n-1) - 85*a(n-2) + 225*a(n-3) - 274*a(n-4) + 120*a(n-5) for n>5.
%F A281774 G.f.: 6*x^3*(1 - 3*x + 10*x^2) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)).
%F A281774 (End)
%t A281774 LinearRecurrence[{15,-85,225,-274,120},{0,0,0,6,72,630},30] (* _Harvey P. Dale_, Oct 22 2018 *)
%o A281774 (PARI) a(n) = 3!*stirling(n, 3, 2) + 3*4!*stirling(n, 4, 2)/2 + 5!*stirling(n, 5, 2) \\ _Colin Barker_, Jan 30 2017
%o A281774 (PARI) concat(vector(3), Vec(6*x^3*(1 - 3*x + 10*x^2) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)) + O(x^30))) \\ _Colin Barker_, Jan 30 2017
%Y A281774 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 A281774 nonn,easy
%O A281774 0,4
%A A281774 Submitted on behalf of Moussa Benoumhani by _Geoffrey Critzer_, Jan 29 2017