cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A057277 Triangle T(n,k) of number of digraphs with a source on n unlabeled nodes with k arcs, k=0..n*(n-1).

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 2, 4, 4, 1, 1, 0, 0, 0, 4, 16, 34, 46, 38, 27, 13, 5, 1, 1, 0, 0, 0, 0, 9, 56, 229, 573, 1058, 1448, 1653, 1487, 1153, 707, 379, 154, 61, 16, 5, 1, 1, 0, 0, 0, 0, 0, 20, 198, 1218, 5089, 15596, 37302, 72776, 119531, 168233, 205923, 220337, 207147, 170965, 124099, 78811, 43861, 21209, 8951, 3242, 1043, 288, 76, 17, 5, 1, 1
Offset: 1

Views

Author

Vladeta Jovovic, Goran Kilibarda, Sep 14 2000

Keywords

Examples

			Triangle begins:
  [1],
  [0, 1, 1],
  [0, 0, 2, 4, 4, 1, 1],
  [0, 0, 0, 4, 16, 34, 46, 38, 27, 13, 5, 1, 1],
  ....
The number of digraphs with a source on 3 unlabeled nodes is 12 = 2+4+4+1+1.

Links

Crossrefs

Row sums give A051421.
Column sums give A350907.
The labeled version is A057274.
Cf. A057270, A057276, A057278, A057279, A350794, A350797.

Programs

Extensions

Terms a(46) and beyond from Andrew Howroyd, Jan 21 2022

A350360 Number of unlabeled digraphs with n nodes containing a global sink (or source).

Original entry on oeis.org

1, 1, 5, 60, 2126, 236560, 86140208, 105190967552, 442114599155408, 6536225731179398016, 345635717436525206325760, 66213119317905480992415271936, 46409685828045501628276172471067136, 119963222885004355352870426935849790038016
Offset: 1

Views

Author

Jim Snyder-Grant, Dec 26 2021

Keywords

Comments

A global sink is a node that has out-degree zero and to which all other nodes have a directed path.
A global source is a node that has in-degree zero and has a directed path to all other nodes. A digraph with a global source, transposed, is a digraph with a global sink.

Examples

			For n=3, 5 digraph edge-sets: (vertex 0 is the single global sink)
  {10,21,20}
  {21,10}
  {21,12,10}
  {21,12,10,20}
  {20,10}

Links

Crossrefs

The labeled version is A350792.
Row sums of A350797.
Cf. A051421, A049531.
Cf. A049512, A350415, A350794, A350798.

Programs

  • PARI
    \\ See PARI link in A350794 for program code.
A350360seq(15) \\ Andrew Howroyd, Jan 21 2022
  • Sage
    # A simple but slow way is to start from all digraphs and filter
# This code can get to n=5
# The linked C code was used to get to n=7
def one_global_sink(g):
    if (g.out_degree().count(0) != 1): return False;
    s = g.out_degree().index(0)
    return [g.distance(v,s) for v in g.vertices()].count(Infinity) == 0
[len([g for g in digraphs(n) if one_global_sink(g)]) for n in (0..5)]

Extensions

Terms a(8) and beyond from Andrew Howroyd, Jan 21 2022

A350795 Triangle read by rows: T(n,k) is the number of digraphs on n unlabeled nodes with k arcs and a global source and sink, n >= 1, k = 0..max(1,n-1)*(n-2)+1.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 6, 8, 4, 1, 0, 0, 0, 0, 1, 16, 70, 140, 159, 113, 53, 17, 4, 1, 0, 0, 0, 0, 0, 1, 33, 313, 1439, 3941, 7297, 9750, 9840, 7717, 4788, 2377, 946, 309, 80, 18, 4, 1, 0, 0, 0, 0, 0, 0, 1, 58, 998, 8447, 43269, 152135, 396011
Offset: 1

Views

Author

Andrew Howroyd, Jan 21 2022

Keywords

Examples

			Triangle begins:
  [1] 1;
  [2] 0, 1;
  [3] 0, 0, 1, 1;
  [4] 0, 0, 0, 1, 6,  8,  4,   1;
  [5] 0, 0, 0, 0, 1, 16, 70, 140, 159, 113, 53, 17, 4, 1;
  ...

Links

Crossrefs

Row sums are A350794.
Column sums are A350796.
The labeled version is A350791.
Cf. A057277, A057278, A350797.

Programs

  • PARI
    \\ See PARI link in A350794 for program code.
{ my(A=A350795triang(5)); for(n=1, #A, print(A[n])) }

A350798 Number of unlabeled digraphs with n arcs and a global source (or sink).

Original entry on oeis.org

1, 1, 2, 6, 21, 84, 387, 1953, 10802, 64267, 408489, 2750686, 19518841, 145236878, 1128884559, 9135566309, 76758391212, 667993213393, 6008434514996, 55755719348210, 532896254763646, 5238211003916685, 52885470406233727, 547751388283907208, 5813693205313803535, 63170000832869927871
Offset: 0

Views

Author

Andrew Howroyd, Jan 21 2022

Keywords

Crossrefs

Column sums of A350797.
Cf. A350360, A350752, A350796, A350906, A350907.

Programs

A350793 Triangle read by rows: T(n,k) is the number of digraphs on n labeled nodes with k arcs and a global source (or sink), n >= 1, k = 0..(n-1)^2.

Original entry on oeis.org

1, 0, 2, 0, 0, 9, 12, 3, 0, 0, 0, 64, 252, 396, 320, 144, 36, 4, 0, 0, 0, 0, 625, 4860, 17060, 35900, 50775, 51300, 38340, 21540, 9075, 2800, 600, 80, 5, 0, 0, 0, 0, 0, 7776, 99720, 603720, 2300310, 6206730, 12654384, 20310840, 26385240, 28273620, 25302960
Offset: 1

Views

Author

Andrew Howroyd, Jan 17 2022

Keywords

Examples

			Triangle begins:
  [1] 1;
  [2] 0, 2;
  [3] 0, 0, 9, 12, 3;
  [4] 0, 0, 0, 64, 252, 396, 320, 144, 36, 4;
  ...

Links

Crossrefs

Row sums are A350792.
The leading diagonal is A000169.
The unlabeled version is A350797.
Cf. A057274, A350791.

Programs

  • PARI
    InitiallyV(n, e=2)={my(v=vector(n)); for(n=1, n, v[n] = n*e^((n-1)^2) - sum(k=1, n-1, binomial(n,k)*e^((n-2)*(n-k))*v[k])); v}
row(n)={Vecrev(InitiallyV(n, 1+'y)[n])}
{ for(n=1, 5, print(row(n))) }
Showing 1-5 of 5 results.

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