A057274 Triangle T(n,k) of the number of digraphs with a source on n labeled nodes with k arcs, k=0..n*(n-1).

1, 0, 2, 1, 0, 0, 9, 20, 15, 6, 1, 0, 0, 0, 64, 330, 720, 914, 792, 495, 220, 66, 12, 1, 0, 0, 0, 0, 625, 5804, 24560, 63940, 117310, 164260, 183716, 167780, 125955, 77520, 38760, 15504, 4845, 1140, 190, 20, 1

Offset: 1

Author

Vladeta Jovovic, Goran Kilibarda, Sep 14 2000

Keywords

• nonn
• tabf

Examples

			Triangle begins:
  1;
  0, 2, 1;
  0, 0, 9, 20,  15,   6,   1;
  0, 0, 0, 64, 330, 720, 914, 792, 495, 220, 66, 12, 1;
  ...
The number of digraphs with a source on 3 labeled nodes is the sum of the terms in row 3, i.e., 0+0+9+20+15+6+1 = 51 = A003028(3).
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..2680 (rows 1..20)
• V. Jovovic and G. Kilibarda, Enumeration of labeled quasi-initially connected digraphs, Discrete Math., 224 (2000), 151-163.

Crossrefs

Row sums give A003028.

The unlabeled version is A057277.

Cf. A057271, A057272, A057273, A057275, A062735.

Programs

PARI

\\ See A057273 for Strong.
Lambda(t, nn, e=2)={my(v=vector(1+nn)); for(n=0, nn, v[1+n] = e^(n*(n+t-1)) - sum(k=0, n-1, binomial(n,k)*e^((n-1)*(n-k))*v[1+k])); v}
Initially(n, e=2)={my(s=Strong(n, e), v=vector(n)); for(k=1, n, my(u=Lambda(k, n-k, e)); for(i=k, n, v[i] += binomial(i,k)*u[1+i-k]*s[k])); v }
row(n)={ Vecrev(Initially(n, 1+'y)[n]) }
{ for(n=1, 5, print(row(n))) } \\ Andrew Howroyd, Jan 11 2022