%I #14 Sep 30 2023 21:46:34
%S 1,1,1,3,7,3,25,85,84,25,543,2335,3579,2322,543,29281,152101,310020,
%T 309725,151835,29281,3781503,23139487,58538763,78349050,58514700,
%U 23128233,3781503,1138779265,8051910805,24318772884,40667112045,40664902810,24315521720,8050866418,1138779265
%N Triangular array read by rows: T(n,k) is the number of Boolean relation matrices such that all of the blocks of its Frobenius normal form are 0blocks or 1blocks and that have exactly k 1blocks on the diagonal, n>=0, 0<=k<=n.
%C A 1(0) block is such that every entry in the block is 1(0).
%C Conjecture: lim_{n > oo} T(n,k)/T(n,nk) = 1.
%H D. A. Gregory, S. Kirkland, and N. J. Pullman, <a href="https://doi.org/10.1016/00243795(93)90323G">Power convergent Boolean matrices</a>, Linear Algebra and its Applications, Volume 179, 15 January 1993, Pages 105117.
%F T(n,0) = T(n,n) = A003024(n).
%F E.g.f.: D(y(exp(x)1)+x) where D(x) is the e.g.f. for A003024.
%e Triangle begins ...
%e 1;
%e 1, 1;
%e 3, 7, 3;
%e 25, 85, 84, 25;
%e 543, 2335, 3579, 2322, 543;
%e 29281, 152101, 310020, 309725, 151835, 29281;
%e 3781503, 23139487, 58538763, 78349050, 58514700, 23128233, 3781503;
%e ...
%t nn = 6; B[n_] := 2^Binomial[n, 2] n!; dags=Select[Import["https://oeis.org/A003024/b003024.txt", "Table"],
%t Length@# == 2 &][[All, 2]]; d[x_] := Total[dags Table[x^i/i!, {i, 0, 40}]];
%t Map[Select[#, # > 0 &] &,Table[n!, {n, 0, nn}] CoefficientList[
%t Series[d[y (Exp[x]  1) + x], {x, 0, nn}], {x, y}]] // Grid
%Y Cf. A365593 (row sums), A003024.
%K nonn,tabl
%O 0,4
%A _Geoffrey Critzer_, Sep 30 2023
