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A057273 Triangle T(n,k) of the number of strongly connected digraphs on n labeled nodes and with k arcs, k=0..n*(n-1).

%I A057273 #24 Jan 11 2022 19:53:24
%S A057273 1,0,0,1,0,0,0,2,9,6,1,0,0,0,0,6,84,316,492,417,212,66,12,1,0,0,0,0,0,
%T A057273 24,720,6440,26875,65280,105566,122580,106825,71700,37540,15344,4835,
%U A057273 1140,190,20,1,0,0,0,0,0,0,120,6480,107850,868830,4188696,13715940
%N A057273 Triangle T(n,k) of the number of strongly connected digraphs on n labeled nodes and with k arcs, k=0..n*(n-1).
%D A057273 Archer, K., Gessel, I. M., Graves, C., & Liang, X. (2020). Counting acyclic and strong digraphs by descents. Discrete Mathematics, 343(11), 112041. See Table 2.
%H A057273 Andrew Howroyd, <a href="/A057273/b057273.txt">Table of n, a(n) for n = 1..2680</a> (rows 1..20)
%H A057273 Kassie Archer, Ira M. Gessel, Christina Graves, and Xuming Liang, <a href="https://arxiv.org/abs/1909.01550">Counting acyclic and strong digraphs by descents</a>, arXiv:1909.01550 [math.CO], 20 Mar 2020. See Table 2.
%e A057273 Triangle begins:
%e A057273   [1] 1;
%e A057273   [2] 0,0,1;
%e A057273   [3] 0,0,0,2,9,6,1;
%e A057273   [4] 0,0,0,0,6,84,316,492,417,212,66,12,1;
%e A057273   ...
%e A057273 Number of strongly connected digraphs on 3 labeled nodes is 18 = 2+9+6+1.
%o A057273 (PARI)
%o A057273 B(nn, e=2)={my(v=vector(nn)); for(n=1, nn, v[n] = e^(n*(n-1)) - sum(k=1, n-1, binomial(n,k)*e^((n-1)*(n-k))*v[k])); v}
%o A057273 Strong(n, e=2)={my(u=B(n, e), v=vector(n)); v[1]=1; for(n=2, #v, v[n] = u[n] + sum(j=1, n-1, binomial(n-1, j-1)*u[n-j]*v[j])); v}
%o A057273 row(n)={ Vecrev(Strong(n, 1+'y)[n]) }
%o A057273 { for(n=1, 5, print(row(n))) } \\ _Andrew Howroyd_, Jan 10 2022
%Y A057273 Row sums give A003030.
%Y A057273 The unlabeled version is A057276.
%Y A057273 Cf. A057271, A057272, A057274, A057275, A062735.
%Y A057273 Cf. A123554, A339590, A339807.
%K A057273 nonn,tabf
%O A057273 1,8
%A A057273 _Vladeta Jovovic_, Goran Kilibarda, Sep 14 2000
%E A057273 Terms a(46) and beyond from _Andrew Howroyd_, Jan 10 2022