A007080 -id:A007080 - cp's OEIS frontend

A176315 Partials sums of A007080.

1, 3, 13, 165, 7901, 1383853, 879285501, 2047199658621, 17660268902020093, 569790880105867285245, 69280640454268889757533693

Offset: 1

Jonathan Vos Post, Apr 15 2010

nonn

Partials sums of number of labeled Eulerian digraphs with n nodes. The subsequence of primes in this partial sum begins: 3, 13, 7901, 1383853, 17660268902020093.

			a(5) = 1 + 2 + 10 + 152 + 7736 = 7901 is prime.

Cf. A007080.

a(n) = SUM[i=1..n] A007080(i).

A229865 Number of n X n 0..1 arrays with corresponding row and column sums equal.

Original entry on oeis.org

1, 2, 8, 80, 2432, 247552, 88060928, 112371410944, 523858015518720, 9041009511609073664, 583447777113052431515648, 141885584718620229407228821504, 130832005909904417592540055577034752, 459749137931232137234615429529864283095040, 6182706200522446492946534924719926752508110700544

Offset: 0

R. H. Hardin, Oct 01 2013

nonn

Also known as labeled Eulerian digraphs allowing loops. - Brendan McKay, May 12 2019

			Some solutions for n=4:
  0 0 0 1     0 0 1 0     0 0 0 1     0 0 1 0     0 0 1 1
  0 1 0 0     1 0 0 0     1 0 1 0     0 0 1 1     1 0 0 1
  0 0 0 1     0 1 0 0     0 1 0 1     0 1 1 1     1 1 1 0
  1 0 1 0     0 0 0 1     0 1 1 0     1 1 0 0     0 1 1 1
From _Gus Wiseman_, Jun 22 2019: (Start)
The a(3) = 8 Eulerian digraph edge-sets:
  {}
  {11}
  {22}
  {11,22}
  {12,21}
  {11,12,21}
  {12,21,22}
  {11,12,21,22}
(End)

Mohammad Behzad Kang and Andrew Salch, The mod p cohomology of the Morava stabilizer group at large primes, arXiv:2410.24171 [math.AT], 2024. See p. 46.

Column 1 of A229870.
The unlabeled version is A308111.
Cf. A000595, A002416, A002720, A007080.
Cf. A326209, A326237, A326251, A326252, A326253.

Table[Length[Select[Subsets[Tuples[Range[n],2]],Sort[First/@#]==Sort[Last/@#]&]],{n,4}] (* Gus Wiseman, Jun 22 2019 *)

a(n) = 2^n * A007080(n). - Andrew Howroyd, Sep 11 2019

a(0)=1 prepended by Alois P. Heinz, May 14 2019

Terms a(11) and beyond from Andrew Howroyd, Sep 11 2019

A058338 Number of digraphs with indegree = outdegree at each vertex, or Eulerian digraphs (including disconnected graphs) with n nodes.

Original entry on oeis.org

1, 1, 2, 5, 18, 111, 2292, 181519, 51294836, 48814342597, 157166848652408, 1736132851915020181, 66689407510956897981608, 9012860249531358132125181323, 4328664025050045497712238253179872, 7451990930474005836163802713084971814275, 46329549163592383403451764167315165130121820112

Offset: 0

N. J. A. Sloane

Every regular tournament (A096368) is a Eulerian digraph. Similar methods may be used to compute terms of this sequence. - Andrew Howroyd, Apr 12 2020

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 219 (but there is an error).
Ronald C. Read, email to N. J. A. Sloane, 28 August, 2000.

Cf. A007080 (labeled), A058337 (connected), A096368, A308161, A308111.

A058337 = Cases[Import["https://oeis.org/A058337/b058337.txt", "Table"], {, }][[All, 2]];
(* EulerTransform is defined in A005195 *)
EulerTransform[Rest @ A058337] (* Jean-François Alcover, Feb 23 2020, upated Mar 18 2020 *)

Euler transform of A058337.

a(7) added using A058337 by Falk Hüffner, Dec 03 2015
a(8) and a(9) added using A058337 by Brendan McKay, May 05 2019
Terms a(10) and beyond from Andrew Howroyd, Apr 12 2020

A229417 T(n,k) = number of n X n 0..k zero-diagonal arrays with corresponding row and column sums equal.

Original entry on oeis.org

1, 1, 2, 1, 3, 10, 1, 4, 45, 152, 1, 5, 136, 4743, 7736, 1, 6, 325, 59008, 3801411, 1375952, 1, 7, 666, 426425, 345706336, 23938685973, 877901648, 1, 8, 1225, 2164680, 11782824375, 28256240134144, 1215663478473627, 2046320373120, 1, 9, 2080

Offset: 1

R. H. Hardin, Sep 22 2013

Table starts
.........1................1....................1................1............1
.........2................3....................4................5............6
........10...............45..................136..............325..........666
.......152.............4743................59008...........426425......2164680
......7736..........3801411............345706336......11782824375.213067487016
...1375952......23938685973.......28256240134144.7093199984236625
.877901648.1215663478473627.33097994593655140864

			Some solutions for n=4 k=4
..0..0..2..0....0..1..0..4....0..0..1..3....0..1..1..4....0..1..1..0
..1..0..2..1....2..0..4..0....1..0..2..3....4..0..2..3....0..0..1..2
..1..2..0..4....2..4..0..2....2..3..0..1....1..4..0..1....0..0..0..4
..0..2..3..0....1..1..4..0....1..3..3..0....1..4..3..0....2..2..2..0

R. H. Hardin, Table of n, a(n) for n = 1..43

Columns 1..3 are A007080, A229415, A229416.
Rows 3..6 are A037270(n+1), A229418, A229419, A229420.
Cf. A229870.

Empirical for row n:
n=1: a(n) = 1
n=2: a(n) = n + 1
n=3: a(n) = (1/2)*n^4 + 2*n^3 + (7/2)*n^2 + 3*n + 1
n=4: [polynomial of degree 9]
Row n is an Ehrhart polynomial of degree (n-1)^2 for the polytope of x(i,j), i,j = 1..n for j <> i, with 0 <= x(i,j) <= 1 and Sum_i x(i,j) = Sum_i x(j,i). - Robert Israel, Mar 30 2023
T(n,k) = A229870(n,k) / (k + 1)^n. - Andrew Howroyd, Mar 30 2023

A054955 Number of labeled Eulerian digraphs with n nodes and an even number of edges.

Original entry on oeis.org

1, 2, 8, 88, 3928, 688336, 438953344, 1023160206720, 8829110851362176, 284886609918484446976, 34640035331694391965082624, 15970703846423879100652101753856, 28060860469435555251136195653129640960, 188681219498365676664628141013181402003089408

Offset: 1

N. J. A. Sloane, May 24 2000

V. A. Liskovets, Some easily derivable sequences, J. Integer Sequences, 3 (2000), #00.2.2.

Cf. A007080, A054956.

A007080 = Cases[Import["https://oeis.org/A007080/b007080.txt", "Table"], {, }][[All, 2]];
a[n_] := (A007080[[n]] + n!)/2;
Array[a, 14] (* Jean-François Alcover, Aug 27 2019 *)

a(n) = (A007080(n) + n!)/2. - Andrew Howroyd, Sep 10 2018

More terms from Vladeta Jovovic, Mar 05 2001

a(12)-a(14) from Andrew Howroyd, Sep 10 2018

A054956 Number of labeled Eulerian digraphs with n nodes and an odd number of edges.

Original entry on oeis.org

0, 0, 2, 64, 3808, 687616, 438948304, 1023160166400, 8829110850999296, 284886609918480818176, 34640035331694391925165824, 15970703846423879100651622752256, 28060860469435555251136195646902620160, 188681219498365676664628141013181314824798208

Offset: 1

N. J. A. Sloane, May 24 2000

V. A. Liskovets, Some easily derivable sequences, J. Integer Sequences, 3 (2000), #00.2.2.

Cf. A007080, A054955.

A007080 = Cases[Import["https://oeis.org/A007080/b007080.txt", "Table"], {, }][[All, 2]];
a[n_] := (A007080[[n]] - n!)/2;
Array[a, 14] (* Jean-François Alcover, Aug 27 2019 *)

a(n) = (A007080(n) - n!)/2. - Andrew Howroyd, Sep 10 2018

More terms from Vladeta Jovovic, Mar 05 2001

a(12)-a(14) from Andrew Howroyd, Sep 10 2018

A054957 Number of labeled connected Eulerian digraphs with n nodes.

Original entry on oeis.org

1, 1, 6, 118, 7000, 1329496, 868255024, 2039295163312, 17639804273910144, 569596637165777524096, 69273803156588266525129984, 31940576331984155835757999795456, 56121305700570450429977355284230355968, 377361653292638108206937443603065676106341376

Offset: 1

N. J. A. Sloane, May 24 2000

Alois P. Heinz, Table of n, a(n) for n = 1..16
V. A. Liskovets, Some easily derivable sequences, J. Integer Sequences, 3 (2000), #00.2.2.

Cf. A058338, A054955, A007080 (EXP-transform), A054959, A054956, A058337, A054958.

a(n) = A054958(n) + A054959(n). - R. J. Mathar, Apr 13 2017

More terms from Vladeta Jovovic, Mar 06 2001

More terms from Alois P. Heinz, Apr 14 2017

A054958 Number of labeled connected Eulerian digraphs with n nodes and an even number of edges.

Original entry on oeis.org

1, 1, 4, 62, 3512, 664808, 434127872, 1019647584176, 8819902136975232, 284798318582888943488, 34636901578294133264379392, 15970288165992077917879019856128, 28060652850285225214988677642354678784, 188680826646319054103468721801532841166681088

Offset: 1

N. J. A. Sloane, May 24 2000

V. A. Liskovets, Some easily derivable sequences, J. Integer Sequences, 3 (2000), #00.2.2.

Cf. A058338, A054955, A007080, A054959, A054956, A058337, A054957.

A054957 = Cases[Import["https://oeis.org/A054957/b054957.txt", "Table"], {, }][[All, 2]];
a[n_] := (A054957[[n]] + (n - 1)!)/2;
Array[a, 14] (* Jean-François Alcover, Sep 13 2019 *)

a(n) = (A054957 + (n-1)!)/2. - Andrew Howroyd, Sep 10 2018

More terms from Vladeta Jovovic, Mar 06 2001

a(12)-a(14) from Andrew Howroyd, Sep 10 2018

A054959 Number of labeled connected Eulerian digraphs with n nodes and an odd number of edges.

Original entry on oeis.org

0, 0, 2, 56, 3488, 664688, 434127152, 1019647579136, 8819902136934912, 284798318582888580608, 34636901578294133260750592, 15970288165992077917878979939328, 28060652850285225214988677641875677184, 188680826646319054103468721801532834939660288

Offset: 1

N. J. A. Sloane, May 24 2000

V. A. Liskovets, Some easily derivable sequences, J. Integer Sequences, 3 (2000), #00.2.2.

Cf. A058338, A054955, A007080, A054956, A058337, A054958, A054957.

A054957 = Cases[Import["https://oeis.org/A054957/b054957.txt", "Table"], {, }][[All, 2]];
a[n_] := (A054957[[n]] - (n - 1)!)/2;
Array[a, 14] (* Jean-François Alcover, Sep 13 2019 *)

a(n) = (A054957(n) - (n-1)!)/2. - Andrew Howroyd, Sep 10 2018

More terms from Vladeta Jovovic, Mar 06 2001

a(12)-a(14) from Andrew Howroyd, Sep 10 2018

A380592 Number of ways that a European soccer league tournament with n teams can complete with all teams having the same number of points.

Original entry on oeis.org

1, 3, 27, 1083, 296081, 696779523, 16503494334993, 3439079361325736243

Offset: 1

Ruediger Jehn, Jan 27 2025

Teams play each other twice for a total of M = n*(n-1) matches.
A victory is awarded 3 points, a draw 1 point and a defeat 0 points.
The total number of possible match outcomes is 3^M = A053764(n) and a(n) is how many of them result in all teams finishing with the same points score.
If all matches were randomly assigned a result, the probability that all teams would end up with the same number of points is a(n)/A053764(n), which in a typical league of 18 or 20 teams is very small.
A007080(n) is the number of ways if there are no draws.

			We denote the vector (r1, r2 ... r_M) with r_i in {0, 1, 3} as a possible sequence of match results. Then a(2) = 3: (0, 0) - both teams lose their home game and have 3 points at the end, (1,1) - both matches end with a draw and both teams have 2 points, (3,3) - both teams win their home game and have 3 points.

Ruediger Jehn, Julia brute-force program
Rüdiger Jehn, Kester Habermann, and Misha Lavrov, Number of ways that a football league can complete with all teams having the same number of points, arXiv:2503.14509 [math.GM], 2025.
Ruediger Jehn, Julia program based on fast algorithm

Cf. A053764, A007080.

cp's OEIS Frontend

A176315 Partials sums of A007080.

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A229865 Number of n X n 0..1 arrays with corresponding row and column sums equal.

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A058338 Number of digraphs with indegree = outdegree at each vertex, or Eulerian digraphs (including disconnected graphs) with n nodes.

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A229417 T(n,k) = number of n X n 0..k zero-diagonal arrays with corresponding row and column sums equal.

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A054955 Number of labeled Eulerian digraphs with n nodes and an even number of edges.

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Mathematica

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A054956 Number of labeled Eulerian digraphs with n nodes and an odd number of edges.

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Author

Keywords

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Programs

Mathematica

Formula

Extensions

A054957 Number of labeled connected Eulerian digraphs with n nodes.

Views

Author

Keywords

Links

Crossrefs

Formula

Extensions

A054958 Number of labeled connected Eulerian digraphs with n nodes and an even number of edges.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A054959 Number of labeled connected Eulerian digraphs with n nodes and an odd number of edges.