A005142 Number of connected bipartite graphs with n nodes. (Formerly M2501)

1, 1, 1, 1, 3, 5, 17, 44, 182, 730, 4032, 25598, 212780, 2241730, 31193324, 575252112, 14218209962, 472740425319, 21208887576786, 1286099113807999, 105567921675718772, 11743905783670560579, 1772771666309380358809, 363526952035325887859823, 101386021137641794979558045

Offset

0,5

Comments

Also, the number of unlabeled connected bicolored graphs having n nodes; the color classes may be interchanged. - Robert W. Robinson

Also, for n>1, number of connected triangle-free graphs on n nodes with chromatic number 2. - Keith Briggs, Mar 21 2006 (cf. A116079).

Also, first diagonal of triangle in A126736.

EULER transform of [1, 1, 1, 3, 5, 17, ...] is A033995 [1, 2, 3, 7, 13, ...]. - Michael Somos, May 13 2019

References

R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.

R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.

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

Links

Andrew Howroyd, Table of n, a(n) for n = 0..50

Jean-François Alcover, Mathematica program.

Keith M. Briggs, Combinatorial Graph Theory, 2024.

CombOS - Combinatorial Object Server, Generate graphs.

Phil Hanlon, The enumeration of bipartite graphs, Disc. Math. 28 (1979), 49-57.

Harrison J. Piehowski, The Lean Number of a Hypergraph, arXiv:2607.14015 [math.CO], 2026. See p. 6 (following Lemma 2.3).

Peter Steinbach, Field Guide to Simple Graphs, Vol. 1, Part 17 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively).

Kai Wang, An Efficient Algorithm to Generate all Labeled Triangle-free Graphs with a given Graphical Degree Sequence, arXiv:2601.15943 [math.CO], 2026. See p. 5.

Eric Weisstein's World of Mathematics, Bipartite Graph..

Eric Weisstein's World of Mathematics, n-Chromatic Graph.

Eric Weisstein's World of Mathematics, n-Colorable Graph.

Jonathan Wurtz and Danylo Lykov, The fixed angle conjecture for QAOA on regular MaxCut graphs, arXiv:2107.00677 [quant-ph], 2021.

Formula

a(2*n+1) = A318870(2*n+1)/2, a(2*n) = (a(n) + A318869(n) + A318870(2*n) - A318870(n))/2. - Andrew Howroyd, Sep 04 2018

Mathematica

(* See the links section. *)

Crossrefs

Cf. A033995, A116079, A318869, A318870.

Sequence in context: A001572 A236458 A131342 * A165452 A106063 A215106

Adjacent sequences: A005139 A005140 A005141 * A005143 A005144 A005145

Keyword

nonn,nice,changed

Author

N. J. A. Sloane

Extensions

More terms from Ronald C. Read

a(0)=1 prepended by Max Alekseyev, Jun 24 2013

Terms a(21) and beyond from Andrew Howroyd, Sep 04 2018

Status

approved