A350909 Triangle read by rows: T(n,k) is the number of weakly connected acyclic digraphs on n labeled nodes with k arcs, k=0..n*(n-1).

1, 0, 2, 0, 0, 12, 6, 0, 0, 0, 128, 186, 108, 24, 0, 0, 0, 0, 2000, 5640, 7840, 6540, 3330, 960, 120, 0, 0, 0, 0, 0, 41472, 189480, 456720, 730830, 832370, 690300, 416160, 178230, 51480, 9000, 720, 0, 0, 0, 0, 0, 0, 1075648, 7178640, 26035800, 65339820

Offset: 1

Author

Andrew Howroyd, Jan 29 2022

Keywords

• nonn
• tabf

Examples

			Triangle begins:
  [1] 1;
  [2] 0, 2;
  [3] 0, 0, 12,   6;
  [4] 0, 0,  0, 128,  186,  108,   24;
  [5] 0, 0,  0,   0, 2000, 5640, 7840, 6540, 3330, 960, 120;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1350 (rows 1..20)

Crossrefs

Row sums are A082402.

Leading diagonal is A097629.

The unlabeled version is A350449.

Cf. A057273, A062735, A081064.

Programs

PARI

G(n)={my(v=vector(n+1)); v[1]=1; for(n=1, n, v[n+1]=sum(k=1, n, -(-1)^k*(1+y)^(k*(n-k))*v[n-k+1]/k!))/n!; Ser(v)}
row(n)={Vecrev(n!*polcoef(log(G(n)), n))}
{ for(n=1, 6, print(row(n))) }