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A370208 Triangular array read by rows. T(n,k) is the number of idempotent binary relations on [n] having no proper power primitive (A360718) with exactly k irreflexive points.

%I A370208 #29 Feb 12 2024 12:44:38
%S A370208 1,1,1,3,6,13,39,87,348,24,841,4205,480,11643,69858,9420,240,227893,
%T A370208 1595251,206640,9240,6285807,50286456,5389552,299040,3360,243593041,
%U A370208 2192337369,172041408,9848160,211680
%N A370208 Triangular array read by rows.  T(n,k) is the number of idempotent binary relations on [n] having no proper power primitive (A360718) with exactly k irreflexive points.
%H A370208 David Rosenblatt, <a href="http://dx.doi.org/10.6028/jres.067B.020">On the graphs of finite Boolean relation matrices</a>, Journal of Research, National Bureau of Standards, Vol 67B No. 4 Oct-Dec 1963.
%F A370208 E.g.f.: 2(exp(y*x*c'(x)/2)-1)*exp(c(x))*exp(x) + exp(c(x))*(y*x*exp(x) + exp(x)) where c(x) is the e.g.f. for A002031.
%e A370208  Triangle begins
%e A370208       1;
%e A370208       1,       1;
%e A370208       3,       6;
%e A370208      13,      39;
%e A370208      87,     348,     24;
%e A370208     841,    4205,    480;
%e A370208   11643,   69858,   9420,  240;
%e A370208  227893, 1595251, 206640, 9240;
%e A370208  ...
%t A370208 nn = 9; A[x_] := Sum[x^n/n! Exp[(2^n - 1) x], {n, 0, nn}];
%t A370208 c[x_] := Log[A[x]] - x; Map[Select[#, # > 0 &] &,
%t A370208  Range[0, nn]! CoefficientList[
%t A370208    Series[2 (Exp[ y x D[c[ x], x]/2] - 1) Exp[c[x]] Exp[ x] +
%t A370208      Exp[c[ x]] (y x Exp[  x] + Exp[ x]), {x, 0, nn}], {x, y}]]
%Y A370208 Cf. A360718 (row sums), A001831 (column k=0), A360743 (T(n,0) + T(n,1) ), A151817 (T(2n,n) for n>=2), A002031.
%K A370208 nonn,tabf
%O A370208 0,4
%A A370208 _Geoffrey Critzer_, Feb 11 2024