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A370464 Triangular array read by rows. T(n,k) is the number of binary relations R on [n] such that the unique idempotent in {R^i:i>=1} contains exactly k non-arcless strongly connected components, n>=0, 0<=k<=n.

%I A370464 #12 Feb 20 2024 10:37:28
%S A370464 1,1,1,3,9,4,25,277,162,48,543,38409,18136,6912,1536,29281,23169481,
%T A370464 7195590,2346000,691200,122880
%N A370464 Triangular array read by rows.  T(n,k) is the number of binary relations R on [n] such that the unique idempotent in {R^i:i>=1} contains exactly k non-arcless strongly connected components, n>=0, 0<=k<=n.
%H A370464 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 A370464 Sum_{n>=1} Sum_{k=1..n} T(n,k)*y^k*x^n/(n!*2^binomial(n,2)) = 1/(E(x) @ exp(- (x + 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 A370464 Triangle begins ...
%e A370464      1;
%e A370464      1,        1;
%e A370464      3,        9,       4;
%e A370464     25,      277,     162,      48;
%e A370464    543,    38409,   18136,    6912,   1536;
%e A370464  29281, 23169481, 7195590, 2346000, 691200, 122880;
%e A370464  ...
%t A370464 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 A370464     4! + (18077431 y + 4815 y^2 + 230 y^3 + 60 y^4 + 24 y^5) x^5/5! ;
%t A370464 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[Series[1/ggf[Exp[-(x + s[x, y])]], {x, 0, nn}], {x, y}]]
%Y A370464 Cf. A002416 (row sums), A003024 (column k=0), A011266 (main diagonal), A370385.
%K A370464 nonn,more,tabl
%O A370464 0,4
%A A370464 _Geoffrey Critzer_, Feb 19 2024