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A370203 Triangular array read by rows. T(n,k) is the number of binary relations on [n] that have exactly k accessible points, n>=0, 0<=k<=n.

%I A370203 #13 Feb 17 2024 13:19:13
%S A370203 1,1,1,3,6,7,25,75,159,253,543,2172,6354,17004,39463,29281,146405,
%T A370203 532130,1841650,6808765,24196201,3781503,22689018,97165485,395729820,
%U A370203 1801073385,9917482698,56481554827
%N A370203 Triangular array read by rows.  T(n,k) is the number of binary relations on [n] that have exactly k accessible points, n>=0, 0<=k<=n.
%C A370203 Let x be in [n]. Then x is accessible by the binary relation R if (x,x) is in R^j for some j>=1. In other words, x is accesible by R if (x,x) is in the transitive closure of R. See Schwarz link.
%H A370203 E. de Panafieu and S. Dovgal, <a href="https://arxiv.org/abs/1903.09454">Symbolic method and directed graph enumeration</a>, arXiv:1903.09454 [math.CO], 2019.
%H A370203 S. Schwarz, <a href="http://dx.doi.org/10.21136/CMJ.1970.100989">On the semigroup of binary relations on a finite set</a>, Czechoslovak Mathematical Journal, 1970.
%F A370203 Sum_{n>=0} T(n,k)*y^k*x^n/(n!*2^binomial(n,2)) = 1/(E(x) @ exp(-(s(2yx)-yx + x))) where E(x) = Sum_{n>=0} x^n/(n!*2^binomial(n,2)), s(x) = Sum_{n>=0} A003030(n)x^n/n! and @ is the exponential Hadamard product (see Panafieu and Dovgal).
%e A370203 Triangle begins:
%e A370203      1;
%e A370203      1,      1;
%e A370203      3,      6,      7;
%e A370203     25,     75,    159,     253;
%e A370203    543,   2172,   6354,   17004,   39463;
%e A370203  29281, 146405, 532130, 1841650, 6808765, 24196201;
%e A370203 ...
%t A370203 nn = 6; B[n_] := n! 2^Binomial[n, 2]; strong = Select[Import["https://oeis.org/A003030/b003030.txt", "Table"], Length@# == 2 &][[All, 2]];s[x_] := Total[strong Table[x^i/i!, {i, 1, 58}]];
%t A370203 ggf[egf_] := Normal[Series[egf, {x, 0, nn}]] /. Table[x^i -> x^i/2^Binomial[i, 2], {i, 0, nn}]; Map[Select[#, # > 0 &] &, Table[B[n], {n, 0, nn}] CoefficientList[Series[1/ggf[Exp[-(s[2 y x] - y x + x)]], {x, 0, nn}], {x, y}]]
%Y A370203 Cf. A002416 (row sums), A003024 (column k = 0), A366866 (main diagonal), A003030.
%K A370203 nonn,tabl
%O A370203 0,4
%A A370203 _Geoffrey Critzer_, Feb 11 2024