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A281775
Number of distinct topologies on an n-set that have exactly 7 open sets.
8
0, 0, 0, 0, 54, 780, 7830, 67620, 535374, 3992940, 28483110, 196316340, 1317106494, 8650141500, 55853351190, 355770438660, 2241509994414, 13998294536460, 86795899256070, 535048203626580, 3282628800655134, 20061393719417820, 122212221633141750
OFFSET
0,5
LINKS
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 (21,-175,735,-1624,1764,-720).
FORMULA
a(n) = 9/4*4! Stirling2(n, 4) + 2*5! Stirling2(n, 5) + 6! Stirling2(n, 6).
From Colin Barker, Jan 30 2017: (Start)
G.f.: 6*x^4*(9 - 59*x + 150*x^2)/((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)).
a(n) = 21*a(n-1) - 175*a(n-2) + 735*a(n-3) - 1624*a(n-4) + 1764*a(n-5) - 720*a(n-6) for n>6.
a(n) = -5 + 17*2^(n-1) - 3^(2+n) + 29*4^(n-1) - 4*5^n + 6^n for n>0. (End)
PROG
(PARI) a(n) = 9*4!*stirling(n, 4, 2)/4 + 2*5!*stirling(n, 5, 2) + 6!*stirling(n, 6, 2) \\ Colin Barker, Jan 30 2017
(PARI) concat(vector(4), Vec(6*x^4*(9 - 59*x + 150*x^2) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*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, A281774, A028244, A281775, A281776, A281777, A281778, A281779, A281780.
Sequence in context: A232829 A368539 A093972 * A344272 A297865 A298132
KEYWORD
nonn,easy
AUTHOR
Submitted on behalf of Moussa Benoumhani by Geoffrey Critzer, Jan 29 2017
STATUS
approved