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A355783 Triangular array read by rows. T(n,k) is the number of labeled transitive relations on [n] that have exactly k symmetric points.

%I A355783 #10 Mar 06 2023 02:57:47
%S A355783 1,2,0,12,0,1,152,0,18,1,3504,0,456,24,10,135392,0,17520,760,600,31,
%T A355783 8321472,0,1015440,35040,40560,2316,361,784621952,0,87375456,2369360,
%U A355783 3615360,185556,52682,2164,110521185024,0,10984707328,233001216,441616000,19052992,7723408,384992,32663
%N A355783 Triangular array read by rows.  T(n,k) is the number of labeled transitive relations on [n] that have exactly k symmetric points.
%C A355783 Let R be a binary relation on [n].  Then x in [n] is a symmetric point of R if there is a y in [n] with x != y and both (x,y),(y,x) in R.
%F A355783 E.g.f.: A(exp(y*x) - 1 - y*x + 2*x) where A(x) is the e.g.f. for A001035.
%e A355783        1,
%e A355783        2, 0,
%e A355783       12, 0,     1,
%e A355783      152, 0,    18,   1,
%e A355783     3504, 0,   456,  24,  10,
%e A355783   135392, 0, 17520, 760, 600, 31
%t A355783 nn = 18; A001035 = Cases[Import["https://oeis.org/A001035/b001035.txt",
%t A355783     "Table"], {_, _}][[All, 2]]; A[x_] = Sum[A001035[[n + 1]] x^n/n!, {n, 0, nn}];
%t A355783 Table[Take[(Range[0, nn]! CoefficientList[Series[A[Exp[y x] - 1 - y x + x + x], {x, 0, nn}], {x,y}])[[i]], i], {i, 1, nn}] // Grid
%Y A355783 Cf. A280202 (main diagonal), A085628 (column k=0), A006905 (row sums).
%K A355783 nonn,tabl
%O A355783 0,2
%A A355783 _Geoffrey Critzer_, Jul 16 2022