A350734 -id:A350734 - cp's OEIS frontend

A350732 Triangle read by rows: T(n,k) is the number of weakly connected oriented graphs on n labeled nodes with k arcs, n >= 0, k=0..n*(n-1)/2.

1, 0, 2, 0, 0, 12, 8, 0, 0, 0, 128, 240, 192, 64, 0, 0, 0, 0, 2000, 7104, 13120, 15360, 11520, 5120, 1024, 0, 0, 0, 0, 0, 41472, 234240, 729600, 1578240, 2531840, 3068928, 2795520, 1863680, 860160, 245760, 32768

Offset: 1

Andrew Howroyd, Jan 11 2022

			Triangle begins:
  [1] 1;
  [2] 0, 2;
  [3] 0, 0, 12,   8;
  [4] 0, 0,  0, 128,  240,  192,    64;
  [5] 0, 0,  0,   0, 2000, 7104, 13120, 15360, 11520, 5120, 1024;
  ...

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

Row sums are A054941.
The leading diagonal is A097629.
The unlabeled version is A350734.
Cf. A062735 (digraphs), A350731 (strongly connected).

row(n)={Vecrev(n!*polcoef(1 + log(sum(k=0, n, (1+2*y)^(k*(k-1)/2)*x^k/k!, O(x*x^n))), n))}
{ for(n=1, 5, print(row(n))) }

A350750 Triangle read by rows: T(n,k) is the number of strongly connected oriented graphs on n unlabeled nodes with k arcs, n >= 1, k = 0..n*(n-1)/2.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 1, 5, 18, 27, 19, 6, 0, 0, 0, 0, 0, 0, 1, 8, 80, 333, 765, 1122, 1049, 622, 217, 35, 0, 0, 0, 0, 0, 0, 0, 1, 12, 221, 1875, 8971, 28449, 63845, 105556, 130935, 122607, 85926, 43868, 15506, 3403, 353

Offset: 1

Andrew Howroyd, Jan 13 2022

			Triangle begins:
  [1] 1;
  [2] 0, 0;
  [3] 0, 0, 0, 1;
  [4] 0, 0, 0, 0, 1, 2, 1;
  [5] 0, 0, 0, 0, 0, 1, 5, 18, 27,  19,   6;
  [6] 0, 0, 0, 0, 0, 0, 1,  8, 80, 333, 765, 1122, 1049, 622, 217, 35;
  ...

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

Row sums are A350489.
Column sums are A350751.
The labeled version is A350731.
Cf. A057276 (digraphs), A350733, A350734.

\\ See PARI link in A350489 for program code.
{ my(A=A350750rows(7)); for(n=1, #A, print(A[n])) }

A350733 Triangle read by rows: T(n,k) is the number of oriented graphs on n unlabeled nodes with k arcs, n >= 0, k = 0..n*(n-1)/2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 4, 10, 12, 10, 4, 1, 1, 4, 13, 41, 78, 131, 144, 107, 50, 12, 1, 1, 4, 14, 55, 187, 539, 1292, 2500, 3817, 4512, 4112, 2740, 1274, 376, 56, 1, 1, 4, 14, 58, 240, 1009, 3643, 11815, 32538, 76145, 149724, 247329, 340364, 387834, 361450, 271177, 159872, 71320, 22690, 4604, 456

Offset: 0

Andrew Howroyd, Jan 13 2022

			Triangle begins:
  [0] 1;
  [1] 1;
  [2] 1, 1;
  [3] 1, 1, 3,  2;
  [4] 1, 1, 4, 10, 12, 10,   4;
  [5] 1, 1, 4, 13, 41, 78, 131, 144, 107, 50, 12;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1350 (rows 0..20)

Row sums are A001174.

Cf. A350734.

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))}
row(n)={my(s=0); forpart(p=n, s+=permcount(p)*edges(p, i->1+2*x^i)); Vecrev(s/n!)}
{ for(n=0, 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])) }

A350915 Number of weakly connected oriented graphs with n arcs.

Original entry on oeis.org

1, 1, 3, 10, 39, 169, 876, 4834, 29316, 189054, 1294382, 9321232, 70326820, 553433559, 4528840412, 38432156859, 337454775045, 3059843449398, 28602687303185, 275222034228537, 2722343346822614, 27647618196693537, 287970349621911635, 3073082817450997700, 33568654163238906968

Offset: 0

Andrew Howroyd, Jan 29 2022

nonn

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

Row sums of A350914.
Column sums of A350734.
Cf. A053454, A086345.

\\ See A350734 for G, InvEulerMTS.
seq(n)=Vec(subst(Pol(InvEulerMTS(sum(i=0, n, G(i, y+O(y^n))*x^i, O(x*x^n)))), x, 1))

cp's OEIS Frontend

A350732 Triangle read by rows: T(n,k) is the number of weakly connected oriented graphs on n labeled nodes with k arcs, n >= 0, k=0..n*(n-1)/2.

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A350750 Triangle read by rows: T(n,k) is the number of strongly connected oriented graphs on n unlabeled nodes with k arcs, n >= 1, k = 0..n*(n-1)/2.

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A350733 Triangle read by rows: T(n,k) is the number of oriented graphs on n unlabeled nodes with k arcs, n >= 0, k = 0..n*(n-1)/2.

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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.

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PARI

A350915 Number of weakly connected oriented graphs with n arcs.

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PARI