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 A281773 Number of distinct topologies on an n-set that have exactly 4 open sets. 9
 0, 0, 1, 9, 43, 165, 571, 1869, 5923, 18405, 56491, 172029, 521203, 1573845, 4742011, 14266989, 42882883, 128812485, 386765131, 1160950749, 3484162963, 10455110325, 31370573851, 94122207309, 282387593443, 847204723365, 2541698056171, 7625261940669 (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 (6,-11,6). FORMULA a(n) = A000392(n+1) + 3*A000392(n). E.g.f.: (exp(x)-1)^3 + (exp(x)-1)^2/2!. From Colin Barker, Jan 30 2017: (Start) G.f.: x^2*(1 + 3*x)/((1 - x)*(1 - 2*x)*(1 - 3*x)). a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) for n>3. a(n) = 2 - 5*2^(n-1) + 3^n for n>0. (End) EXAMPLE a(3) = 9 because we have: {{}, {c}, {a,b}, {a,b,c}} with 3 labelings and {{}, {c}, {b,c}, {a,b,c}} with 6 labelings. MATHEMATICA CoefficientList[Series[x^2*(1 + 3 x)/((1 - x) (1 - 2 x) (1 - 3 x)), {x, 0, 27}], x] (* Michael De Vlieger, Jan 21 2018 *) PROG (PARI) a(n) = stirling(n, 2, 2) + 3!*stirling(n, 3, 2) \\ Colin Barker, Jan 30 2017 (PARI) concat(vector(2), Vec(x^2*(1 + 3*x) / ((1 - x)*(1 - 2*x)*(1 - 3*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. Partial sums are given in A298564. Sequence in context: A244869 A259181 A330088 * A220676 A110125 A221751 Adjacent sequences: A281770 A281771 A281772 * A281774 A281775 A281776 KEYWORD nonn,easy AUTHOR Submitted on behalf of Moussa Benoumhani by Geoffrey Critzer, Jan 29 2017 STATUS approved

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