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A370385 Triangular array read by rows. T(n,k) is the number of binary relations R on [n] such that the unique idempotent relation in {R^i:i>=1} is a quasi-order containing exactly k strongly connected components.

%I A370385 #25 Feb 20 2024 10:37:11
%S A370385 1,1,3,4,139,66,48,25575,9280,3072,1536,18077431,4498530,1174800,
%T A370385 322560,122880
%N A370385 Triangular array read by rows. T(n,k) is the number of binary relations R on [n] such that the unique idempotent relation in {R^i:i>=1} is a quasi-order containing exactly k strongly connected components.
%H A370385 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.
%F A370385 Sum_{n>=1} Sum_{k=1..n} T(n,k)*y^k*x^n/(n!*2^binomial(n,2)) = 1/(E(x) @ exp(-s(x,y))) where E(x) = Sum_{n>=0} x^n/(n!*2^binomial(n,2)) and @ is the exponential Hadamard product (see Panafieu and Dovgal) and s(x,y) is the e.g.f. for A367948.
%e A370385 Triangle begins:
%e A370385         1;
%e A370385         1;
%e A370385         3,       4;
%e A370385       139,      66,      48;
%e A370385     25575,    9280,    3072,   1536;
%e A370385  18077431, 4498530, 1174800, 322560, 122880;
%e A370385  ...
%t A370385 nn = 5; B[n_] := n! 2^Binomial[n, 2]; s[x_, y_] := y x + (3 y + y^2) x^2/2! + (139 y + 3 y^2 + 2 y^3) x^3/3! + (25575 y + 103 y^2 + 12 y^3 + 6 y^4) x^4/
%t A370385     4! + (18077431 y + 4815 y^2 + 230 y^3 + 60 y^4 + 24 y^5) x^5/5! ;
%t A370385 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[
%t A370385    Series[1/ggf[Exp[-s[x, y]]], {x, 0, nn}], {x, y}]]
%Y A370385 Cf. A366866 (row sums), A070322 (column k=1), A011266 (main diagonal), A367948, A247231, A370464.
%K A370385 nonn,more,tabf
%O A370385 1,3
%A A370385 _Geoffrey Critzer_, Feb 18 2024