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 A281774 Number of distinct topologies on an n-set with exactly 6 open sets. 9
 0, 0, 0, 6, 72, 630, 4680, 31206, 193032, 1131990, 6386760, 35025606, 188061192, 993760950, 5187840840, 26831095206, 137770476552, 703455087510, 3576115150920, 18117222864006, 91536570671112, 461496288791670, 2322770028381000, 11675109032796006 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 LINKS Colin Barker, Table of n, a(n) for n = 0..1000 Moussa Benoumhani, The Number of Topologies on a Finite Set, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.6. Index entries for linear recurrences with constant coefficients, signature (15,-85,225,-274,120). FORMULA a(n) = 3! StirlingS2(n, 3) + 3/2*4! StirlingS2(n, 4) + 5! StirlingS2(n, 5). From Colin Barker, Jan 30 2017: (Start) a(n) = 2 - 2^(2+n) - 7*2^(2*n-1) + 5*3^n + 5^n for n>5. 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. 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)). (End) MATHEMATICA LinearRecurrence[{15, -85, 225, -274, 120}, {0, 0, 0, 6, 72, 630}, 30] (* Harvey P. Dale, Oct 22 2018 *) PROG (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 (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 CROSSREFS 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. Sequence in context: A151719 A118313 A283095 * A036292 A061690 A133678 Adjacent sequences:  A281771 A281772 A281773 * A281775 A281776 A281777 KEYWORD nonn,easy AUTHOR Submitted on behalf of Moussa Benoumhani by Geoffrey Critzer, Jan 29 2017 STATUS approved

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Last modified April 1 02:24 EDT 2020. Contains 333153 sequences. (Running on oeis4.)