%I #41 Apr 20 2023 11:26:38
%S 1,0,1,1,0,3,1,9,0,19,10,12,114,0,219,31,300,190,2190,0,4231,361,1158,
%T 10140,4380,63465,0,130023,2164,26341,46389,451920,148085,2730483,0,
%U 6129859,32663,192496,1930852,2381624,27601000,7281288,171636052,0,431723379
%N Triangle read by rows: T(n,k) = number of topologies on an n-set X such that there are exactly k elements in X that are topologically distinguishable, n >= 0, 0 <= k <= n.
%C 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.
%C 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
%H Alois P. Heinz, <a href="/A280192/b280192.txt">Rows n = 0..18, flattened</a>
%H B. M. Schein, <a href="https://doi.org/10.3792/pja/1195520400">A construction for idempotent binary relations</a>, Proc. Japan Acad., Vol. 46, No. 3 (1970), pp. 246-247.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Topological_indistinguishability">Topological indistinguishability</a>.
%F E.g.f.: A(exp(x) - 1 - x + y*x) where A(x) is the e.g.f. for A001035.
%F Sum_{k=0..n} T(n,k)*2^k = A006905(n). - _Geoffrey Critzer_, Apr 18 2023
%e Triangle begins:
%e 1;
%e 0, 1;
%e 1, 0, 3;
%e 1, 9, 0, 19;
%e 10, 12, 114, 0, 219;
%e 31, 300, 190, 2190, 0, 4231;
%e 361, 1158, 10140, 4380, 63465, 0, 130023;
%e 2164, 26341, 46389, 451920, 148085, 2730483, 0, 6129859;
%e ...
%t A001035 = Cases[Import["https://oeis.org/A001035/b001035.txt", "Table"], {_, _}][[All, 2]];
%t lg = Length[A001035];
%t A[x_] = Sum[A001035[[n + 1]] x^n/n!, {n, 0, lg - 1}];
%t 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 *)
%Y Right border gives A001035.
%Y Row sums give A000798.
%Y Column k=0 gives A280202.
%Y Cf. A006905.
%K nonn,tabl
%O 0,6
%A _Geoffrey Critzer_, Dec 28 2016