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.

1, 0, 2, 0, 0, 6, 6, 0, 0, 0, 24, 132, 180, 84, 12, 0, 0, 0, 0, 120, 1800, 8000, 16160, 18180, 12580, 5560, 1560, 260, 20, 0, 0, 0, 0, 0, 720, 22320, 214800, 999450, 2764650, 5125380, 6844380, 6882150, 5355750, 3277200, 1586520, 605370, 179250, 39900, 6300, 630, 30

Offset: 1

Author

Andrew Howroyd, Jan 16 2022

Keywords

• nonn
• tabf

Examples

			Triangle begins:
  [1] 1;
  [2] 0, 2;
  [3] 0, 0, 6, 6;
  [4] 0, 0, 0, 24, 132, 180, 84, 12;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..2319 (rows 1..20)
• R. W. Robinson, Counting digraphs with restrictions on the strong components, Combinatorics and Graph Theory '95 (T.-H. Ku, ed.), World Scientific, Singapore (1995), 343-354.

Crossrefs

Row sums are A350790.

The unlabeled version is A350795.

Cf. A057271, A350793.

Programs

PARI

\\ Following Eqn 21 in the Robinson reference.
Z(p,f)={my(n=serprec(p,x)); serconvol(p, sum(k=0, n-1, x^k*f(k), O(x^n)))}
G(e,p)={Z(p, k->1/e^(k*(k-1)/2))}
U(e,p)={Z(p, k->e^(k*(k-1)/2))}
DigraphEgf(n,e)={sum(k=0, n, e^(k*(k-1))*x^k/k!, O(x*x^n) )}
StrongD(n,e=2)={-log(U(e, 1/G(e, DigraphEgf(n, e))))}
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))) ))}
row(n)={Vecrev(InitFinallyV(n, 1+'y)[n]) }
{ for(n=1, 5, print(row(n))) }