A126736 -id:A126736 - cp's OEIS frontend

A005142 Number of connected bipartite graphs with n nodes.

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

N. J. A. Sloane

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 M. 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

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).

Andrew Howroyd, Table of n, a(n) for n = 0..50
Jean-François Alcover, Mathematica program
Keith M. Briggs, Combinatorial Graph Theory
CombOS - Combinatorial Object Server, Generate graphs
P. Hanlon, The enumeration of bipartite graphs, Discrete Math. 28 (1979), 49-57.
Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 17 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
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.

Cf. A033995, A116079, A318869, A318870.

Mathematica
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(* See the links section. *)
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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

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

A084269 Triangle read by rows: T(n,k) is the number of simple connected graphs on n unlabeled nodes having chromatic number k, 1 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 2, 1, 0, 5, 12, 3, 1, 0, 17, 64, 26, 4, 1, 0, 44, 475, 282, 46, 5, 1, 0, 182, 5036, 5009, 809, 74, 6, 1, 0, 730, 80947, 149551, 27794, 1940, 110, 7, 1, 0, 4032, 2010328, 7694428, 1890221, 113272, 4125, 156, 8, 1, 0, 25598, 76115143, 667036310, 248580644, 14545025, 389583, 8040, 212, 9, 1

Offset: 0

Eric W. Weisstein, May 24 2003

			Triangle begins:
  1;
  0,   1;
  0,   1,     1;
  0,   3,     2,      1;
  0,   5,    12,      3,     1;
  0,  17,    64,     26,     4,    1;
  0,  44,   475,    282,    46,    5,   1;
  0, 182,  5036,   5009,   809,   74,   6, 1;
  0, 730, 80947, 149551, 27794, 1940, 110, 7, 1;
  ...

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

Row sums are A001349.
Columns k=3..7 are A126737, A126738, A126739, A126740, A241702.
Partial row sums include A005142, A076322, A076323, A076324, A076325, A076326, A076327, A076328.
Essentially the same table as A126736.
Cf. A084268 (not necessarily connected), A115597.

a(37)-a(66) from Andrew Howroyd, Dec 02 2018

A126737 Number of connected graphs on n nodes with chromatic number 3.

Original entry on oeis.org

0, 0, 1, 2, 12, 64, 475, 5036, 80947, 2010328, 76115143

Offset: 1

N. J. A. Sloane, Feb 16 2007

nonn

Keith M. Briggs, Combinatorial Graph Theory

A diagonal of triangle in A126736.

a(11) from Michael Sollami, Jan 29 2012

A126738 Number of connected graphs on n nodes with chromatic number 4.

Original entry on oeis.org

0, 0, 0, 1, 3, 26, 282, 5009, 149551, 7694428, 667036310

Offset: 1

N. J. A. Sloane, Feb 16 2007

nonn

Keith M. Briggs, Combinatorial Graph Theory

A diagonal of triangle in A126736.

a(11) from Michael Sollami, Jan 29 2012

A126739 Number of connected graphs on n nodes with chromatic number 5.

Original entry on oeis.org

0, 0, 0, 0, 1, 4, 46, 809, 27794, 1890221, 248580644

Offset: 1

N. J. A. Sloane, Feb 16 2007

nonn

Keith M. Briggs, Combinatorial Graph Theory

A diagonal of triangle in A126736.

a(11) from Michael Sollami, Jan 29 2012

A126740 Number of connected graphs on n nodes with chromatic number 6.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 5, 74, 1940, 113272, 14545025

Offset: 1

N. J. A. Sloane, Feb 16 2007

nonn

Keith M. Briggs, Combinatorial Graph Theory

A diagonal of triangle in A126736.

a(11) from Michael Sollami, Jan 29 2012

A241702 Number of connected graphs on n nodes with chromatic number 7.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 6, 110, 4125, 389583

Offset: 1

Travis Hoppe and Anna Petrone, Apr 27 2014

A diagonal of triangle in A126736.

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
T. Hoppe and A. Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
F. Hüffner, tinygraph, software for generating integer sequences based on graph properties.
Eric Weisstein's World of Mathematics, Chromatic Number

a(11) added using tinygraph by Falk Hüffner, Apr 04 2017

cp's OEIS Frontend

A005142 Number of connected bipartite graphs with n nodes.

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A084269 Triangle read by rows: T(n,k) is the number of simple connected graphs on n unlabeled nodes having chromatic number k, 1 <= k <= n.

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A126737 Number of connected graphs on n nodes with chromatic number 3.

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A126738 Number of connected graphs on n nodes with chromatic number 4.

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A126739 Number of connected graphs on n nodes with chromatic number 5.

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A126740 Number of connected graphs on n nodes with chromatic number 6.

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A241702 Number of connected graphs on n nodes with chromatic number 7.

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