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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.
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%I #29 Feb 12 2024 12:44:38

%S 1,1,1,3,6,13,39,87,348,24,841,4205,480,11643,69858,9420,240,227893,

%T 1595251,206640,9240,6285807,50286456,5389552,299040,3360,243593041,

%U 2192337369,172041408,9848160,211680

%N 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 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 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 Triangle begins

%e 1;

%e 1, 1;

%e 3, 6;

%e 13, 39;

%e 87, 348, 24;

%e 841, 4205, 480;

%e 11643, 69858, 9420, 240;

%e 227893, 1595251, 206640, 9240;

%e ...

%t nn = 9; A[x_] := Sum[x^n/n! Exp[(2^n - 1) x], {n, 0, nn}];

%t c[x_] := Log[A[x]] - x; Map[Select[#, # > 0 &] &,

%t Range[0, nn]! CoefficientList[

%t Series[2 (Exp[ y x D[c[ x], x]/2] - 1) Exp[c[x]] Exp[ x] +

%t Exp[c[ x]] (y x Exp[ x] + Exp[ x]), {x, 0, nn}], {x, y}]]

%Y Cf. A360718 (row sums), A001831 (column k=0), A360743 (T(n,0) + T(n,1) ), A151817 (T(2n,n) for n>=2), A002031.

%K nonn,tabf

%O 0,4

%A _Geoffrey Critzer_, Feb 11 2024