A281777 Number of distinct topologies on an n-set that have exactly 9 open sets.

0, 0, 0, 0, 20, 800, 14260, 189280, 2181060, 23241120, 235737620, 2308206560, 21979728100, 204477713440, 1864504348980, 16707856095840, 147469451067140, 1284607771225760, 11063319237792340, 94343562846289120, 797685042851814180, 6694943490279586080

Offset: 0

Author

Submitted on behalf of Moussa Benoumhani by Geoffrey Critzer, Jan 29 2017

Keywords

• nonn
• easy

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 (36,-546,4536,-22449,67284,-118124,109584,-40320).

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.

Programs

Mathematica

LinearRecurrence[{36,-546,4536,-22449,67284,-118124,109584,-40320},{0,0,0,0,20,800,14260,189280,2181060},30] (* Harvey P. Dale, Aug 19 2020 *)

PARI

concat(vector(4), Vec(20*x^4*(1 + 4*x - 181*x^2 + 1100*x^3 - 1344*x^4) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)*(1 - 8*x)) + O(x^30))) \\ Colin Barker, Jan 30 2017

Formula

a(n) = 5/6*4! Stirling2(n, 4) + 5*5! Stirling2(n, 5) + 11/2*6! Stirling2(n, 6) + 3*7! Stirling2(n, 7) + 8! Stirling2(n, 8).

G.f.: 20*x^4*(1 + 4*x - 181*x^2 + 1100*x^3 - 1344*x^4) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)*(1 - 8*x)). - Colin Barker, Jan 30 2017