A350915 -id:A350915 - cp's OEIS frontend

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

Andrew Howroyd, Jan 13 2022

			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;
  ...

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

Row sums are A086345.
Column sums are A350915.
Leading diagonal is A000238.
The labeled version is A350732.
Cf. A054733, A350733, A350750, A350914 (transpose).

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))) }

A350914 Triangle read by rows: T(n,k) is the number of unlabeled weakly connected oriented graphs with n arcs and k vertices, k = 1..n+1.

Original entry on oeis.org

1, 0, 1, 0, 0, 3, 0, 0, 2, 8, 0, 0, 0, 12, 27, 0, 0, 0, 10, 68, 91, 0, 0, 0, 4, 127, 395, 350, 0, 0, 0, 0, 144, 1144, 2170, 1376, 0, 0, 0, 0, 107, 2393, 9139, 11934, 5743, 0, 0, 0, 0, 50, 3767, 28606, 67104, 64892, 24635, 0, 0, 0, 0, 12, 4500, 71583, 288331, 468702, 352286, 108968

Offset: 0

Andrew Howroyd, Jan 29 2022

			Triangle begins:
  1;
  0, 1;
  0, 0, 3;
  0, 0, 2,  8;
  0, 0, 0, 12,  27;
  0, 0, 0, 10,  68,   91;
  0, 0, 0,  4, 127,  395,  350;
  0, 0, 0,  0, 144, 1144, 2170, 1376;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Row sums are A350915.
Column sums are A086345.
Cf. A350734 (transpose), A350789 (digraphs).

\\ See A350734 for G, InvEulerMTS.
T(n)={my(p=InvEulerMTS(sum(i=0, n, G(i, y+O(y^n))*x^i, O(x*x^n)))); vector(n, n, Vec(O(x^n)+polcoef(p,n-1,y)/x, -n))}
{ my(A=T(10)); for(n=1, #A, print(A[n])) }

cp's OEIS Frontend

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.

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