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 1-4 of 4 results.

A054548 Triangular array giving number of labeled graphs on n unisolated nodes and k=0...n*(n-1)/2 edges.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 3, 1, 0, 0, 3, 16, 15, 6, 1, 0, 0, 0, 30, 135, 222, 205, 120, 45, 10, 1, 0, 0, 0, 15, 330, 1581, 3760, 5715, 6165, 4945, 2997, 1365, 455, 105, 15, 1, 0, 0, 0, 0, 315, 4410, 23604, 73755, 159390, 259105, 331716, 343161, 290745, 202755, 116175
Offset: 0

Views

Author

Vladeta Jovovic, Apr 09 2000

Keywords

Examples

			From _Gus Wiseman_, Feb 14 2024: (Start)
Triangle begins:
   1
   0
   0   1
   0   0   3   1
   0   0   3  16  15   6   1
   0   0   0  30 135 222 205 120  45  10   1
Row n = 4 counts the following graphs:
  .  .  12-34  12-13-14  12-13-14-23  12-13-14-23-24  12-13-14-23-24-34
        13-24  12-13-24  12-13-14-24  12-13-14-23-34
        14-23  12-13-34  12-13-14-34  12-13-14-24-34
               12-14-23  12-13-23-24  12-13-23-24-34
               12-14-34  12-13-23-34  12-14-23-24-34
               12-23-24  12-13-24-34  13-14-23-24-34
               12-23-34  12-14-23-24
               12-24-34  12-14-23-34
               13-14-23  12-14-24-34
               13-14-24  12-23-24-34
               13-23-24  13-14-23-24
               13-23-34  13-14-23-34
               13-24-34  13-14-24-34
               14-23-24  13-23-24-34
               14-23-34  14-23-24-34
               14-24-34
(End)

References

  • F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973, Page 29, Exercise 1.4.

Links

Crossrefs

Row sums give A006129. Cf. A054547.
The connected case is A062734, with loops A369195.
This is the covering case of A084546.
Column sums are A121251, with loops A173219.
The version with loops is A369199, row sums A322661.
The unlabeled version is A370167, row sums A002494.
A006125 counts simple graphs; also loop-graphs if shifted left.
Cf. A000666, A006649, A062740, A066383, A121252, A133686, A322700, A368597, A369196, A369197.

Programs

  • Mathematica
    nn=5; s=Sum[(1+y)^Binomial[n,2]  x^n/n!, {n,0,nn}]; Range[0,nn]! CoefficientList[Series[ s Exp[-x], {x,0,nn}], {x,y}] //Grid  (* returns triangle indexed at n = 0, Geoffrey Critzer, Oct 07 2012 *)
Table[Length[Select[Subsets[Subsets[Range[n],{2}],{k}],Union@@#==Range[n]&]],{n,0,5},{k,0,Binomial[n,2]}] (* Gus Wiseman, Feb 14 2024 *)

Formula

T(n, k) = Sum_{i=0..n} (-1)^(n-i)*C(n, i)*C(C(i, 2), k), k=0...n*(n-1)/2.
E.g.f.: exp(-x)*Sum_{n>=0} (1 + y)^C(n,2)*x^n/n!. - Geoffrey Critzer, Oct 07 2012

Extensions

a(0) prepended by Gus Wiseman, Feb 14 2024

A053418 Number of unlabeled directed graphs with n arcs and no isolated vertices.

Original entry on oeis.org

1, 1, 5, 17, 80, 365, 1981, 11222, 69511, 455663, 3169244, 23170347, 177513359, 1418920570, 11798710013, 101778754655, 908722427531, 8380602471646, 79692654473866, 780142956502644, 7851084073063731, 81120767066417308
Offset: 0

Views

Author

Vladeta Jovovic, Jan 10 2000

Keywords

Links

Crossrefs

The labeled version is A121252.
Column sums of A350908.
Cf. A000273, A000664, A053454, A053598 (by # of nodes).

Formula

Euler transform of A053454. - Andrew Howroyd, Jan 28 2022

Extensions

Edited and extended by Max Alekseyev, Sep 18 2009

A054547 Triangular array giving number of labeled digraphs on n unisolated nodes and k=0..n*(n-1) arcs.

Original entry on oeis.org

0, 0, 2, 1, 0, 0, 12, 20, 15, 6, 1, 0, 0, 12, 140, 435, 768, 920, 792, 495, 220, 66, 12, 1, 0, 0, 0, 240, 2520, 11604, 34150, 73560, 123495, 166860, 184426, 167900, 125965, 77520, 38760, 15504, 4845, 1140, 190, 20, 1
Offset: 1

Views

Author

Vladeta Jovovic, Apr 09 2000

Keywords

Examples

			Triangle T(n,k) begins:
  [0],
  [0,2,1],
  [0,0,12,20,15,6,1],
  [0,0,12,140,435,768,920,792,495,220,66,12,1],
  ...

Links

Crossrefs

Row sums are A054545.
Column sums are A121252.
The unlabeled version is A350908.
Cf. A054548 (graphs), A062735, A123554.

Programs

  • PARI
    row(n) = {Vecrev(sum(i=0, n, (-1)^(n-i)*binomial(n,i)*(1 + 'y)^(i*(i-1))), n*(n-1)+1)}
{ for(n=1, 6, print(row(n))) } \\ Andrew Howroyd, Jan 28 2022

Formula

T(n, k) = Sum_{i=0..n} (-1)^(n-i)*binomial(n, i)*binomial(i*(i-1), k).

A121933 Number of labeled digraphs with n arcs for which every vertex has indegree at least one and outdegree at least one.

Original entry on oeis.org

1, 0, 1, 2, 18, 158, 1788, 23930, 370886, 6527064, 128542420, 2800362536, 66858556196, 1735834171276, 48689118113374, 1467253017578672, 47275138863637080, 1621757692715997136, 59013695834307968254, 2270400832166224741596, 92078072790064946096284
Offset: 0

Views

Author

Vladeta Jovovic, Sep 02 2006

Keywords

Links

Crossrefs

Cf. A121252, A086193 (by # of nodes), A367500 (unlabeled version).

Programs

  • Maple
    n:=20: t:=taylor(sum(sum((-1)^(m-k)*binomial(m,k)*((1+x)^(k-1)-1)^k*((1+x)^k-1)^(m-k),k=0..m),m=0..n),x,n+1): seq(coeff(t,x,m),m=0..n); # Nathaniel Johnston, Apr 28 2011
  • Mathematica
    Flatten[{1,Rest[CoefficientList[Series[Sum[Sum[(-1)^(n-k)*Binomial[n,k]*((1+x)^(k-1)-1)^k*((1+x)^k-1)^(n-k),{k,0,n}],{n,1,20}],{x,0,20}],x]]}] (* Vaclav Kotesovec, May 07 2014 *)

Formula

G.f.: Sum(Sum((-1)^(n-k)*binomial(n,k)*((1+x)^(k-1)-1)^k*((1+x)^k-1)^(n-k),k=0..n),n=0..infinity).
a(n) ~ c * n! / (sqrt(n) * (log(2))^(2*n)), where c = 0.0722246614111436... . - Vaclav Kotesovec, May 07 2014
In closed form, c = 1/(sqrt(Pi*(1-log(2))) * log(2) * 2^(4+log(2)/2)). - Vaclav Kotesovec, May 04 2015
Showing 1-4 of 4 results.

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