A350734 Triangle read by rows: T(n,k) is the number of weakly connected oriented graphs on n unlabeled nodes with k arcs, n >= 1, k = 0..n*(n-1)/2.

1, 0, 1, 0, 0, 3, 2, 0, 0, 0, 8, 12, 10, 4, 0, 0, 0, 0, 27, 68, 127, 144, 107, 50, 12, 0, 0, 0, 0, 0, 91, 395, 1144, 2393, 3767, 4500, 4112, 2740, 1274, 376, 56, 0, 0, 0, 0, 0, 0, 350, 2170, 9139, 28606, 71583, 145600, 244589, 339090, 387458, 361394, 271177, 159872, 71320, 22690, 4604, 456

Offset: 1

Author

Andrew Howroyd, Jan 13 2022

Keywords

• nonn
• tabf

Examples

			Triangle begins:
  [1] 1;
  [2] 0, 1;
  [3] 0, 0, 3, 2;
  [4] 0, 0, 0, 8, 12, 10,   4;
  [5] 0, 0, 0, 0, 27, 68, 127, 144, 107, 50, 12;
  ...
		

Links

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

Crossrefs

Row sums are A086345.

Column sums are A350915.

Leading diagonal is A000238.

The labeled version is A350732.

Cf. A054733, A350733, A350750, A350914 (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)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c-1)\2))}
G(n, x)={my(s=0); forpart(p=n, s+=permcount(p)*edges(p, i->1+2*x^i)); s/n!}
row(n)={Vecrev(polcoef(InvEulerMTS(sum(i=0, n, G(i, y)*x^i, O(x*x^n))), n))}
{ for(n=1, 6, print(row(n))) }