A054547 Triangular array giving number of labeled digraphs on n unisolated nodes and k=0..n*(n-1) arcs.

0, 0, 2, 1, 0, 0, 12, 20, 15, 6, 1, 0, 0, 12, 140, 435, 768, 920, 792, 495, 220, 66, 12, 1, 0, 0, 0, 240, 2520, 11604, 34150, 73560, 123495, 166860, 184426, 167900, 125965, 77520, 38760, 15504, 4845, 1140, 190, 20, 1

Offset: 1

Author

Vladeta Jovovic, Apr 09 2000

Keywords

• easy
• nonn
• tabf

Examples

			Triangle T(n,k) begins:
  [0],
  [0,2,1],
  [0,0,12,20,15,6,1],
  [0,0,12,140,435,768,920,792,495,220,66,12,1],
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..2680 (rows 1..20)

Crossrefs

Row sums are A054545.

Column sums are A121252.

The unlabeled version is A350908.

Cf. A054548 (graphs), A062735, A123554.

Programs

PARI

row(n) = {Vecrev(sum(i=0, n, (-1)^(n-i)*binomial(n,i)*(1 + 'y)^(i*(i-1))), n*(n-1)+1)}
{ for(n=1, 6, print(row(n))) } \\ Andrew Howroyd, Jan 28 2022

Formula

T(n, k) = Sum_{i=0..n} (-1)^(n-i)*binomial(n, i)*binomial(i*(i-1), k).