A057273 -id:A057273 - cp's OEIS frontend

A272582 The number of strongly connected digraphs with n vertices and n+1 edges.

0, 9, 84, 720, 6480, 63000, 665280, 7620480, 94348800, 1257379200, 17962560000, 273988915200, 4446092851200, 76498950528000, 1391365527552000, 26676557107200000, 537799391281152000, 11373816888225792000, 251805357846282240000, 5824367407574876160000

Offset: 2

R. J. Mathar, May 10 2016

Wright also gives the number of strongly connected digraphs with n vertices and n+2 edges, 0, 6, 316, 6440, 107850, 1719060, 27476400, ... (offset 2) in terms of a polynomial of order 5 multiplied by n!. - R. J. Mathar, May 12 2016

Andrew Howroyd, Table of n, a(n) for n = 2..200
E. M. Wright, Formulae for the number of sparsely-edged strong labelled digraphs, Quart. J. Math. 28 (3) (1977) 363-367, Section 3.

A diagonal of A057273.

Table[(n-2)(n+3)n!/4,{n,2,30}] (* Harvey P. Dale, May 23 2017 *)

a(n) = (n-2)*(n+3)*n!/4 \\ Andrew Howroyd, Jan 15 2022

from _future_ import print_function
from sympy import factorial
for n in range(2,500):
   print((int)((n-2)*(n+3)*factorial(n)/4),end=", ")
# Soumil Mandal, May 12 2016

a(n) = (n-2)*(n+3)*n!/4.
E.g.f.: x^3*(3 - 2*x)/(2*(1 - x)^3). - Ilya Gutkovskiy, May 10 2016
D-finite with recurrence -(n+1)*(n-4)*a(n) +(n-1)*(n-3)*(n+2)*a(n-1)=0. - R. J. Mathar, Mar 11 2021

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).

Original entry on oeis.org

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

Andrew Howroyd, Jan 29 2022

			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;
  ...

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

Row sums are A082402.
Leading diagonal is A097629.
The unlabeled version is A350449.
Cf. A057273, A062735, A081064.

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))) }

cp's OEIS Frontend

A272582 The number of strongly connected digraphs with n vertices and n+1 edges.

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