A000664 Number of graphs with n edges. (Formerly M1400 N0545)

1, 1, 2, 5, 11, 26, 68, 177, 497, 1476, 4613, 15216, 52944, 193367, 740226, 2960520, 12334829, 53394755, 239544624, 1111261697, 5320103252, 26237509076, 133087001869, 693339241737, 3705135967663, 20286965943329, 113694201046379, 651571521170323, 3815204365835840, 22806847476040913, 139088381010541237, 864777487052916454

Offset

0,3

Comments

These are simple graphs, unlabeled, with no isolated nodes, but are not necessarily connected.

References

W. Oberschelp, Kombinatorische Anzahlbestimmungen in Relationen, Math. Ann., 174 (1967), 53-78.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 146.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Max Alekseyev, Table of n, a(n) for n = 0..60

Nicolas Borie, The Hopf Algebra of graph invariants, arXiv preprint arXiv:1511.05843 [math.CO], 2015.

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

Fran Herr and Legrand Jones II, Iterated Jump Graphs, arXiv:2205.01796 [math.CO], 2022.

M. L. Stein and P. R. Stein, Enumeration of Linear Graphs and Connected Linear Graphs up to p = 18 Points. Report LA-3775, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Oct 1967

Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Overview of the 11 Parts (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)

Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 1

Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 2

Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 3

Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 4

Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 5

Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 6

Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 7

Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 8

Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 9

Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 10

Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 11

Formula

a(n) = A008406(2*n,n). - Max Alekseyev, Sep 13 2016

Euler transform of A002905 (ignoring A002905(0)). - Franklin T. Adams-Watters Jul 03 2009

Example

n=1: o-o (1)
n=2: o-o o-o, o-o-o (2)
n=3: o-o o-o o-o, o-o-o o-o, o-o-o-o, Y, triangle (5)
n=4: o-o o-o o-o o-o, o-o-o o-o o-o, o-o-o o-o-o, o-o o-o-o-o, o-o Y, o-o triangle,
o-o-o-o-o, >o-o-o, ><, square, triangle with tail (11)

Mathematica

<< Combinatorica`; Table[NumberOfGraphs[2 n, n], {n, 0, 10}] (* Eric W. Weisstein, Oct 30 2017 *)
<< Combinatorica`; Table[Coefficient[GraphPolynomial[2 n, x], x, n], {n, 0, 10}] (* Eric W. Weisstein, Oct 30 2017 *)

Crossrefs

Cf. A002905, A008406, A053418.

Row sums of A275421.

Cf. also A000088, A000055.

Sequence in context: A239311 A319760 A343872 * A364597 A242766 A182580

Adjacent sequences: A000661 A000662 A000663 * A000665 A000666 A000667

Keyword

nonn,nice

Author

N. J. A. Sloane

Extensions

More terms from Vladeta Jovovic, Jan 08 2000, Aug 14 2007

Edited by N. J. A. Sloane, Feb 26 2008

Example for n=2 corrected by Adrian Falcone (falcone(AT)gmail.com), Jan 28 2009

Zero term inserted by Franklin T. Adams-Watters, Jul 03 2009

a(25)-a(26) from Max Alekseyev, Sep 19 2009

a(27)-a(60) from Max Alekseyev, Sep 07 2016

Status

approved