A054733 Triangle of number of (weakly) connected unlabeled digraphs with n nodes and k arcs (n >=2, k >= 1).

1, 1, 0, 3, 4, 4, 1, 1, 0, 0, 8, 22, 37, 47, 38, 27, 13, 5, 1, 1, 0, 0, 0, 27, 108, 326, 667, 1127, 1477, 1665, 1489, 1154, 707, 379, 154, 61, 16, 5, 1, 1, 0, 0, 0, 0, 91, 582, 2432, 7694, 19646, 42148, 77305, 122953, 170315, 206982, 220768, 207301, 171008

Offset: 2

Author

Vladeta Jovovic, Apr 21 2000

Keywords

• easy
• nonn
• tabf

Examples

			1,1;
0,3,4,4,1,1;
0,0,8,22,37,47,38,27,13,5,1,1;
the last batch giving the numbers of connected digraphs with 4 nodes and from 1 to 12 arcs.
		

References

• F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973.

Links

• Andrew Howroyd, Table of n, a(n) for n = 2..2661 (rows 2..20)
• R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 (2017) Table 75.

Crossrefs

Cf. A000238 (leading diagonal), A003085 (row sums), A053454 (column sums), A062735 (labeled).

Cf. A052283 (not necessarily connected), A283753 (another version), A057276 (strongly connected), A350789 (transpose).

Programs

PARI

InvEulerMTS(p)={my(n=serprec(p,x)-1, q=log(p), vars=variables(p)); sum(i=1, n, moebius(i)*substvec(q + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
edges(v, t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i], v[j])); t(v[i]*v[j]/g)^(2*g) )) * prod(i=1, #v, my(c=v[i]); t(c)^(c-1))}
G(n, x)={my(s=0); forpart(p=n, s+=permcount(p)*edges(p, i->1+x^i)); s/n!}
row(n)={Vecrev(polcoef(InvEulerMTS(sum(i=0, n, G(i, y)*x^i, O(x*x^n))), n)/y)}
{ for(n=2, 6, print(row(n))) } \\ Andrew Howroyd, Jan 28 2022