A308161 -id:A308161 - cp's OEIS frontend

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

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

A308111 Isomorphism classes of Eulerian digraphs with n vertices, allowing loops.

Original entry on oeis.org

1, 2, 6, 24, 160, 2512, 129816, 22665792, 13056562208, 24953006054144, 160860329639968800, 3555065836569542246400, 273147301191314006316868352, 73832333258502021627712839197696, 70920540648597652305602460997787710080, 244186544390677638132290202415190606165938176, 3036252267734950687777830287721323374283100639476736

Offset: 0

Brendan McKay, May 11 2019

Eulerian means that for every vertex the in-degree equals the out-degree.

			For n=2 the a(2)=6 solutions are: two non-adjacent vertices with or without loops (3 cases), two vertices with or without loops connected by edges in each direction (3 cases).

Gus Wiseman, Non-isomorphic representatives of the a(3) = 24 Eulerian digraphs with loops.

For labeled digraphs rather than isomorphism classes see A229865.
For isomorphism classes with loops forbidden see A058338.
Cf. A308128 (connected version of this).
Cf. A000273, A000595, A002416, A002720, A308161.

Euler transform of A308128.

Terms a(10) and beyond from Andrew Howroyd, Apr 12 2020

A308239 Number of isomorphism classes of connected Eulerian oriented graphs with n vertices.

Original entry on oeis.org

1, 1, 0, 1, 1, 4, 16, 175, 5274, 434017, 90658149, 48825116761, 68579602126387, 254236892409251687, 2512196983110257299898, 66738846701458593981832314, 4802164152196181961357340585741, 942009012794429918939576183310877477, 506665422212653541096106256584713063500057

Offset: 0

Alois P. Heinz, May 16 2019

Loops and 2-cycles are not permitted. Eulerian means that each vertex has equal in-degree and out-degree.

Cf. A308161.

Inverse Euler transform of A308161.

Terms a(12) and beyond from Andrew Howroyd, Apr 10 2020

cp's OEIS Frontend

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

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A308111 Isomorphism classes of Eulerian digraphs with n vertices, allowing loops.

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A308239 Number of isomorphism classes of connected Eulerian oriented graphs with n vertices.

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