OFFSET
0,6
COMMENTS
T(n,0) = A280202(n) is the number of topologies on an n-set X such that for all x in X there exists a y in X such that x and y have exactly the same neighborhoods.
Equivalently, T(n,k) is the number of labeled quasi-orders R on [n] with exactly k singletons in the equivalence relation R intersect R^(-1), cf. Schein link. - Geoffrey Critzer, Apr 18 2023
LINKS
Alois P. Heinz, Rows n = 0..18, flattened
B. M. Schein, A construction for idempotent binary relations, Proc. Japan Acad., Vol. 46, No. 3 (1970), pp. 246-247.
Wikipedia, Topological indistinguishability.
FORMULA
E.g.f.: A(exp(x) - 1 - x + y*x) where A(x) is the e.g.f. for A001035.
Sum_{k=0..n} T(n,k)*2^k = A006905(n). - Geoffrey Critzer, Apr 18 2023
EXAMPLE
Triangle begins:
1;
0, 1;
1, 0, 3;
1, 9, 0, 19;
10, 12, 114, 0, 219;
31, 300, 190, 2190, 0, 4231;
361, 1158, 10140, 4380, 63465, 0, 130023;
2164, 26341, 46389, 451920, 148085, 2730483, 0, 6129859;
...
MATHEMATICA
lg = Length[A001035];
A[x_] = Sum[A001035[[n + 1]] x^n/n!, {n, 0, lg - 1}];
CoefficientList[#, y]& /@ (CoefficientList[A[Exp[x] - 1 - x + y*x] + O[x]^lg, x]*Range[0, lg - 1]!) // Flatten (* Jean-François Alcover, Jan 01 2020 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Dec 28 2016
STATUS
approved