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A350791 Triangle read by rows: T(n,k) is the number of digraphs on n labeled nodes with k arcs and a global source and sink, n >= 1, k = 0..max(1,n-1)*(n-2)+1.

%I A350791 #12 Jan 22 2022 23:42:58
%S A350791 1,0,2,0,0,6,6,0,0,0,24,132,180,84,12,0,0,0,0,120,1800,8000,16160,
%T A350791 18180,12580,5560,1560,260,20,0,0,0,0,0,720,22320,214800,999450,
%U A350791 2764650,5125380,6844380,6882150,5355750,3277200,1586520,605370,179250,39900,6300,630,30
%N A350791 Triangle read by rows: T(n,k) is the number of digraphs on n labeled nodes with k arcs and a global source and sink, n >= 1, k = 0..max(1,n-1)*(n-2)+1.
%H A350791 Andrew Howroyd, <a href="/A350791/b350791.txt">Table of n, a(n) for n = 1..2319</a> (rows 1..20)
%H A350791 R. W. Robinson, <a href="http://cobweb.cs.uga.edu/~rwr/publications/components.pdf">Counting digraphs with restrictions on the strong components</a>, Combinatorics and Graph Theory '95 (T.-H. Ku, ed.), World Scientific, Singapore (1995), 343-354.
%e A350791 Triangle begins:
%e A350791   [1] 1;
%e A350791   [2] 0, 2;
%e A350791   [3] 0, 0, 6, 6;
%e A350791   [4] 0, 0, 0, 24, 132, 180, 84, 12;
%e A350791   ...
%o A350791 (PARI) \\ Following Eqn 21 in the Robinson reference.
%o A350791 Z(p,f)={my(n=serprec(p,x)); serconvol(p, sum(k=0, n-1, x^k*f(k), O(x^n)))}
%o A350791 G(e,p)={Z(p, k->1/e^(k*(k-1)/2))}
%o A350791 U(e,p)={Z(p, k->e^(k*(k-1)/2))}
%o A350791 DigraphEgf(n,e)={sum(k=0, n, e^(k*(k-1))*x^k/k!, O(x*x^n) )}
%o A350791 StrongD(n,e=2)={-log(U(e, 1/G(e, DigraphEgf(n, e))))}
%o A350791 InitFinallyV(n, e=2)={my(S=StrongD(n, e)); Vec(serlaplace( x - x^2 + exp(S) * U(e, G(e, x*exp(-S))^2*G(e, DigraphEgf(n,e))) ))}
%o A350791 row(n)={Vecrev(InitFinallyV(n, 1+'y)[n]) }
%o A350791 { for(n=1, 5, print(row(n))) }
%Y A350791 Row sums are A350790.
%Y A350791 The unlabeled version is A350795.
%Y A350791 Cf. A057271, A350793.
%K A350791 nonn,tabf
%O A350791 1,3
%A A350791 _Andrew Howroyd_, Jan 16 2022