A350731 -id:A350731 - 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])) }

A350730 Number of strongly connected oriented graphs on n labeled nodes.

Original entry on oeis.org

1, 0, 2, 66, 7998, 2895570, 3015624078, 8890966977354, 74079608267459142, 1754419666770364130730, 119163820122708911990211222, 23431180614718394105521543222866, 13448652672256961901980839022683943838, 22684139279519345808802725789494254587951810

Offset: 1

Andrew Howroyd, Jan 11 2022

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..50

The unlabeled version is A350489.
Row sums of A350731.
Cf. A003030, A054941.

StrongO(14) \\ See A350731 for program code.

A350749 Triangle read by rows: T(n,k) is the number of oriented graphs on n labeled nodes with k arcs, n >= 0, k = 0..n*(n-1)/2.

Original entry on oeis.org

1, 1, 1, 2, 1, 6, 12, 8, 1, 12, 60, 160, 240, 192, 64, 1, 20, 180, 960, 3360, 8064, 13440, 15360, 11520, 5120, 1024, 1, 30, 420, 3640, 21840, 96096, 320320, 823680, 1647360, 2562560, 3075072, 2795520, 1863680, 860160, 245760, 32768

Offset: 0

Andrew Howroyd, Feb 15 2022

			Triangle begins:
  [0] 1;
  [1] 1;
  [2] 1,  2;
  [3] 1,  6,  12,   8;
  [4] 1, 12,  60, 160,  240,  192,    64;
  [5] 1, 20, 180, 960, 3360, 8064, 13440, 15360, 11520, 5120, 1024;
  ...

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

Row sums are A047656.
The unlabeled version is A350733.
Cf. A013609, A350732 (weakly connected), A350731 (strongly connected).

PARI
```
T(n,k) = 2^k * binomial(n*(n-1)/2, k)
```

row(n) = {Vecrev((1+2*y)^(n*(n-1)/2))}
{ for(n=0, 6, print(row(n))) }

T(n,k) = 2^k * binomial(n*(n-1)/2, k) = A013609(n*(n-1)/2, k).

T(n,k) = [y^k] (1+2*y)^(n*(n-1)/2).

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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A350730 Number of strongly connected oriented graphs on n labeled nodes.

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PARI

A350749 Triangle read by rows: T(n,k) is the number of oriented graphs on n labeled nodes with k arcs, n >= 0, k = 0..n*(n-1)/2.

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