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%I #14 Oct 22 2023 16:52:54
%S 1,1,1,4,9,3,51,298,138,25,1831,40815,17853,4494,543,166930,23752151,
%T 7418420,1861755,325895,29281,36681301,55427713806,10701675348,
%U 2105585760,391017795,53021223,3781503
%N Triangular array read by rows. T(n,k) is the number of labeled digraphs (with self loops allowed) on [n] containing exactly k primitive components, n>=0, 0<=k<=n.
%C A primitive component (A070322) is a strongly connected component (A003030) such that the gcd of the lengths of its cycles is 1.
%H 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 Sum_{n>=0} T(n,k)*y^k*x^n/(n!*2^binomial(n,2)) = 1/(E(x) @ exp(-(y*p(x)-1)+ s(2x) - (p(x)-1))) where E(x) = Sum_{n>=0} x^n/(n!*2^binomial(n,2)), p(x) is the e.g.f. for A070322, s(x) is the e.g.f. for A003030 and @ is the exponential Hadamard product (see Panafieu and Dovgal).
%e Triangle begins
%e 1;
%e 1, 1;
%e 4, 9, 3;
%e 51, 298, 138, 25;
%e 1831, 40815, 17853, 4494, 543;
%e ...
%t nn = 6; B[n_] := 2^Binomial[n, 2] n!; strong = Select[Import["https://oeis.org/A003030/b003030.txt", "Table"], Length@# == 2 &][[All, 2]];s[x_] := Total[strong Table[x^i/i!, {i, 1, 58}]]; primitive =
%t Select[Import["https://oeis.org/A070322/b070322.txt", "Table"],
%t Length@# == 2 &][[All, 2]]; pr[x_] := Total[primitive Table[x^i/i!, {i, 0, 6}]];ggf[egf_] := Normal[Series[egf, {x, 0, nn}]] /. Table[x^i ->x^i/2^Binomial[i, 2], {i, 0, nn}];
%t Map[Select[#, # > 0 &] &,Table[B[n], {n, 0, nn}] CoefficientList[Series[1/ggf[Exp[- (y (pr[x] - 1) + s[2 x] - (pr[x] - 1))]], {x,
%t 0, nn}], {x, y}]] // Grid
%Y Cf. A002416 (row sums), A003024 (main diagonal), A070322, A003030, A361269.
%K nonn,tabl
%O 0,4
%A _Geoffrey Critzer_, Oct 22 2023
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