login
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
OFFSET
0,4
LINKS
Moussa Benoumhani, The Number of Topologies on a Finite Set, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.6.
FORMULA
a(n) = 3! Stirling2(n, 3) + 3/2*4! Stirling2(n, 4) + 5! Stirling2(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
KEYWORD
nonn,easy
AUTHOR
Submitted on behalf of Moussa Benoumhani by Geoffrey Critzer, Jan 29 2017
STATUS
approved