A281776 Number of distinct topologies on an n-set that have exactly 8 open sets.

0, 0, 0, 1, 54, 955, 11760, 122941, 1175034, 10595215, 91506420, 763624081, 6194818014, 49084747075, 381338401080, 2914184784421, 21965095364994, 163656285828535, 1207613518375740, 8838842878371961, 64253768864671974, 464416229729871595, 3340518964319750400

Offset: 0

Author

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 (28,-322,1960,-6769,13132,-13068,5040).

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

PARI

concat(vector(3), Vec(x^3*(1 + 26*x - 235*x^2 + 448*x^3 + 2100*x^4) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)) + O(x^30))) \\ Colin Barker, Jan 30 2017

Formula

a(n) = Stirling2(n, 3) + 2*4! Stirling2(n, 4) + 15/4*5! Stirling2(n, 5) + 5/2*6! Stirling2(n, 6) + 7! Stirling2(n, 7).

From Colin Barker, Jan 30 2017: (Start)

a(n) = 13/4 - 19*2^(n-1) + 44*3^(n-1) - 2^(n-1)*3^(2+n) - 57*4^(n-1) + (39*5^n)/4 + 7^n for n>0.

G.f.: x^3*(1 + 26*x - 235*x^2 + 448*x^3 + 2100*x^4) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)).

(End)