A002905 Number of connected graphs with n edges. (Formerly M2486 N0985)

1, 1, 1, 3, 5, 12, 30, 79, 227, 710, 2322, 8071, 29503, 112822, 450141, 1867871, 8037472, 35787667, 164551477, 779945969, 3804967442, 19079312775, 98211456209, 518397621443, 2802993986619, 15510781288250, 87765472487659, 507395402140211, 2994893000122118, 18035546081743772, 110741792670074054, 692894304050453139

Offset

0,4

References

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

G. A. Baker et al., High-temperature expansions for the spin-1/2 Heisenberg model, Phys. Rev., 164 (1967), 800-817.

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.

Mike Cummings and Adam Van Tuyl, The GeometricDecomposability package for Macaulay2, arXiv:2211.02471 [math.AC], 2022.

Anjan Dutta and Hichem Sahbi, Graph Kernels based on High Order Graphlet Parsing and Hashing, arXiv:1803.00425 [cs.CV], 2018.

Gordon Royle, Small graphs

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, Part 1 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)

Eric Weisstein's World of Mathematics, Polynema.

Formula

A000664 and this sequence are an Euler transform pair. - N. J. A. Sloane, Aug 30 2016

Example

a(3) = 3 since the three connected graphs with three edges are a path, a triangle and a "Y".
The first difference between this sequence and A046091 is for n=9 edges where we see K_{3,3}, the well-known "utility graph".

Mathematica

A000664 = Cases[Import["https://oeis.org/A000664/b000664.txt", "Table"], {_, _}][[All, 2]];
(* EulerInvTransform is defined in A022562 *)
Join[{1}, EulerInvTransform[Rest @ A000664]] (* Jean-François Alcover, May 10 2019, updated Mar 17 2020 *)

Crossrefs

Column sums of A054924 or equivalently row sums of A054923.

Cf. A000664, A046091 (for connected planar graphs), A275421 (multisets).

Apart from a(3), same as A003089.

Sequence in context: A066951 A295065 A046091 * A220832 A323270 A087610

Adjacent sequences: A002902 A002903 A002904 * A002906 A002907 A002908

Keyword

nonn,nice

Author

N. J. A. Sloane

Extensions

More terms from Vladeta Jovovic, Jan 12 2000

More terms from Gordon F. Royle, Jun 05 2003

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

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

Status

approved