A033995 -id:A033995 - cp's OEIS frontend

A123408 Duplicate of A033995.

1, 2, 3, 7, 13, 35, 88, 303, 1119, 5479

Offset: 1

dead

A047863 Number of labeled graphs with 2-colored nodes where black nodes are only connected to white nodes and vice versa.

1, 2, 6, 26, 162, 1442, 18306, 330626, 8488962, 309465602, 16011372546, 1174870185986, 122233833963522, 18023122242478082, 3765668654914699266, 1114515608405262434306, 467221312005126294077442, 277362415313453291571118082, 233150477220213193598856331266

Offset: 0

N. J. A. Sloane

Row sums of A111636. - Peter Bala, Sep 30 2012
Column 2 of Table 2 in Read. - Peter Bala, Apr 11 2013
It appears that 5 does not divide a(n), that a(n) is even for n>0, that 3 divides a(2n) for n>0, that 7 divides a(6n+5), and that 13 divides a(12n+3). - Ralf Stephan, May 18 2013

			For n=2, {1,2 black, not connected}, {1,2 white, not connected}, {1 black, 2 white, not connected}, {1 black, 2 white, connected}, {1 white, 2 black, not connected}, {1 white, 2 black, connected}.
G.f. = 1 + 2*x + 6*x^2 + 26*x^3 + 162*x^4 + 1442*x^5 + 18306*x^6 + ...

H. S. Wilf, Generatingfunctionology, Academic Press, NY, 1990, p. 79, Eq. 3.11.2.

T. D. Noe, Table of n, a(n) for n = 0..50
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
S. R. Finch, Bipartite, k-colorable and k-colored graphs
S. R. Finch, Bipartite, k-colorable and k-colored graphs, June 5, 2003. [Cached copy, with permission of the author]
A. Gainer-Dewar and I. M. Gessel, Enumeration of bipartite graphs and bipartite blocks, arXiv:1304.0139 [math.CO], 2013.
D. A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19. [Annotated scanned copy]
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
R. C. Read, The number of k-colored graphs on labelled nodes, Canad. J. Math., 12 (1960), 410-414.
R. P. Stanley, Acyclic orientation of graphs Discrete Math. 5 (1973), 171-178. North Holland Publishing Company.
Martin Svatoš, Peter Jung, Jan Tóth, Yuyi Wang, and Ondřej Kuželka, On Discovering Interesting Combinatorial Integer Sequences, arXiv:2302.04606 [cs.LO], 2023, p. 17.
Eric Weisstein's World of Mathematics, k-Colorable Graph
H. S. Wilf, Generatingfunctionology, 2nd edn., Academic Press, NY, 1994, p. 88, Eq. 3.11.2.

Column k=2 of A322280.
Cf. A135079 (variant).
Cf. A000683, A001831, A002031, A033995, A047864, A052332, A111636, A191371, A223887.

A047863:= func< n | (&+[Binomial(n,k)*2^(k*(n-k)): k in [0..n]]) >;
[A047863(n): n in [0..40]]; // G. C. Greubel, Nov 03 2024

Table[Sum[Binomial[n,k]2^(k(n-k)),{k,0,n}],{n,0,20}] (* Harvey P. Dale, May 09 2012 *)
nmax = 20; CoefficientList[Series[Sum[E^(2^k*x)*x^k/k!, {k, 0, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 05 2019 *)

{a(n)=n!*polcoeff(sum(k=0,n,exp(2^k*x +x*O(x^n))*x^k/k!),n)} \\ Paul D. Hanna, Nov 27 2007

{a(n)=polcoeff(sum(k=0, n, x^k/(1-2^k*x +x*O(x^n))^(k+1)), n)} \\ Paul D. Hanna, Mar 08 2008

N=66; x='x+O('x^N); egf = sum(n=0, N, exp(2^n*x)*x^n/n!);
Vec(serlaplace(egf))  \\ Joerg Arndt, May 04 2013

from sympy import binomial
def a(n): return sum([binomial(n, k)*2**(k*(n - k)) for k in range(n + 1)]) # Indranil Ghosh, Jun 03 2017

def A047863(n): return sum(binomial(n,k)*2^(k*(n-k)) for k in range(n+1))
[A047863(n) for n in range(41)] # G. C. Greubel, Nov 03 2024

a(n) = Sum_{k=0..n} binomial(n, k)*2^(k*(n-k)).
a(n) = 4 * A000683(n) + 2. - Vladeta Jovovic, Feb 02 2000
E.g.f.: Sum_{n>=0} exp(2^n*x)*x^n/n!. - Paul D. Hanna, Nov 27 2007
O.g.f.: Sum_{n>=0} x^n/(1 - 2^n*x)^(n+1). - Paul D. Hanna, Mar 08 2008
From Peter Bala, Apr 11 2013: (Start)
Let E(x) = Sum_{n >= 0} x^n/(n!*2^C(n,2)) = 1 + x + x^2/(2!*2) + x^3/(3!*2^3) + .... Then a generating function is E(x)^2 = 1 + 2*x + 6*x^2/(2!*2) + 26*x^3/(3!*2^3) + .... In general, E(x)^k, k = 1, 2, ..., is a generating function for labeled k-colored graphs (see Stanley). For other examples see A191371 (k = 3) and A223887 (k = 4).
If A(x) = 1 + 2*x + 6*x^2/2! + 26*x^3/3! + ... denotes the e.g.f. for this sequence then sqrt(A(x)) = 1 + x + 2*x^2/2! + 7*x^3/3! + ... is the e.g.f. for A047864, which counts labeled 2-colorable graphs. (End)
a(n) ~ c * 2^(n^2/4+n+1/2)/sqrt(Pi*n), where c = Sum_{k = -infinity..infinity} 2^(-k^2) = EllipticTheta[3, 0, 1/2] = 2.128936827211877... if n is even and c = Sum_{k = -infinity..infinity} 2^(-(k+1/2)^2) = EllipticTheta[2, 0, 1/2] = 2.12893125051302... if n is odd. - Vaclav Kotesovec, Jun 24 2013

Better description from Christian G. Bower, Dec 15 1999

A005142 Number of connected bipartite graphs with n nodes.

Original entry on oeis.org

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
```
(* See the links section. *)
```

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

A006785 Number of triangle-free graphs on n vertices.

Original entry on oeis.org

1, 2, 3, 7, 14, 38, 107, 410, 1897, 12172, 105071, 1262180, 20797002, 467871369, 14232552452, 581460254001, 31720840164950

Offset: 1

N. J. A. Sloane

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

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
CombOS - Combinatorial Object Server, Generate graphs
P. Erdős, D. J. Kleitman, and B. L. Rothschild, Asymptotic enumeration of k_n-free graphs. In Colloquio Internazionale sulle Teorie Combinatorie, (Rome, 1973), Tomo II, Atti dei Convegni Lincei, No. 17, pp. 19-27. Accad. Naz. Lincei, Rome.
Jérôme Kunegis, Jun Sun, and Eiko Yoneki, Guided Graph Generation: Evaluation of Graph Generators in Terms of Network Statistics, and a New Algorithm, arXiv:2303.00635 [cs.SI], 2023, p. 17.
Brendan McKay, Emails to N. J. A. Sloane, 1991
B. D. McKay, Isomorph-free exhaustive generation, J Algorithms, 26 (1998) 306-324.
W. Pu, J. Choi, and E. Amir, Lifted Inference On Transitive Relations, Workshops at the Twenty-Seventh AAAI Conference on Statistical Relational Artificial Intelligence, 2013.
Eric Weisstein's World of Mathematics, Triangle-Free Graph

Cf. A024607.

Row sums of A283417.

Erdős, Kleitman, & Rothschild prove that a(n) = 2^(n^2/4 + o(n^2)) and a(n) = (1 + o(1/n))*A033995(n). - Charles R Greathouse IV, Feb 01 2018

2 more terms (from the McKay paper) from Vladeta Jovovic, May 17 2008

2 more terms from Brendan McKay, Jan 12 2013

A047864 Number of labeled bipartite graphs with n nodes.

Original entry on oeis.org

1, 1, 2, 7, 41, 376, 5177, 103237, 2922446, 116011231, 6433447397, 498234407452, 54007795331921, 8213123246906761, 1756336596363006842, 528975889250504033527, 224688018516023267969441, 134708289561117007261966816

Offset: 0

N. J. A. Sloane

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 406.
H. S. Wilf, Generatingfunctionology, Academic Press, NY, 1990, p. 80, Eq. 3.11.5.

T. D. Noe, Table of n, a(n) for n = 0..50
Vladislav Bína and Jiří Přibil, Note on enumeration of labeled split graphs, Comment. Math. Univ. Carolin. 56,2 (2015) 133-137.
S. R. Finch, Bipartite, k-colorable and k-colored graphs. [Broken link]
S. R. Finch, Bipartite, k-colorable and k-colored graphs, June 5, 2003. [Cached copy, with permission of the author]
Qipeng Kuang, Ondřej Kuželka, Yuanhong Wang, and Yuyi Wang, Bridging Weighted First Order Model Counting and Graph Polynomials, arXiv:2407.11877 [cs.LO], 2024. See p. 32.
Eric Weisstein's World of Mathematics, n-Colorable Graph.
H. S. Wilf, Generatingfunctionology, 2nd edn., Academic Press, NY, 1994, p. 89, Eq. 3.11.5.

Row sums of A117279.
The unlabeled version is A033995.
Cf. A001832, A047863.

nn = 20; a = Sum[Sum[Binomial[n, k] 2^(k (n - k)), {k, 0, n}] x^n/n!, {n, 0, nn}]; Range[0, nn]! CoefficientList[Series[a^(1/2), {x, 0, nn}], x]  (* Geoffrey Critzer, Jan 15 2012 *)

N=18; x='x+O('x^N); Vec(serlaplace(sqrt(sum(n=0, N, exp(2^n*x)*x^n/n!)))) \\ Gheorghe Coserea, Nov 13 2017

E.g.f.: sqrt( e.g.f. for A047863 ).

A084268 Triangle read by rows: T(n,k) is the number of simple graphs on n unlabeled nodes having chromatic number k, 1 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 3, 1, 1, 12, 16, 4, 1, 1, 34, 84, 31, 5, 1, 1, 87, 579, 318, 52, 6, 1, 1, 302, 5721, 5366, 867, 81, 7, 1, 1, 1118, 87381, 155291, 28722, 2028, 118, 8, 1, 1, 5478, 2104349, 7855628, 1919895, 115391, 4251, 165, 9, 1, 1, 32302, 78315231, 675054876, 250530482, 14662562, 393963, 8214, 222, 10, 1

Offset: 1

Eric W. Weisstein, May 24 2003

T(n,1) = T(n,n) = 1 (here we count the empty graph and the complete graph). T(n,n-1) = n-1 (here we count the graphs with clique number equal to n-1). - Geoffrey Critzer, Oct 12 2016

Row sums give A000088. - Joerg Arndt, Oct 13 2016

			Triangle begins:
  1;
  1,    1;
  1,    2,       1;
  1,    6,       3,       1;
  1,   12,      16,       4,       1;
  1,   34,      84,      31,       5,      1;
  1,   87,     579,     318,      52,      6,    1;
  1,  302,    5721,    5366,     867,     81,    7,   1;
  1, 1118,   87381,  155291,   28722,   2028,  118,   8, 1;
  1, 5478, 2104349, 7855628, 1919895, 115391, 4251, 165, 9, 1;
  ...

Keith Briggs, combinatorial graph theory, see entry "number of graphs on n nodes with clique number k".
FindStat - Combinatorial Statistic Finder, The chromatic number of a graph.
Eric Weisstein's World of Mathematics, Chromatic Number
Eric Weisstein's World of Mathematics, n-Chromatic Graph

Columns k=1..9 are A057427, A076278, A076279, A076280, A076281, A076282, A076283, A205567, A205568.
Partial row sums include A033995, A076315, A076316, A076317, A076318, A076319, A076320, A076321.
Row sums are A000088.
Cf. A084269 (connected), A115597 (essentially the same sequence).

# prints triangle with a leading zero in each row
for n in range(1, 8) :
    st = [0 for j in range(n+1)]
    G = graphs(n)
    for g in G :
        st[ g.chromatic_number() ] += 1
    print(st)
# Joerg Arndt, Oct 13 2016

Offset corrected by Joerg Arndt, Oct 13 2016
a(36)-a(55) from Joerg Arndt, Oct 15 2016
a(56)-a(66) from Andrew Howroyd, Dec 02 2018

A076278 Number of 2-chromatic (i.e., chromatic number equals 2) simple graphs on n nodes.

Original entry on oeis.org

0, 1, 2, 6, 12, 34, 87, 302, 1118, 5478, 32302, 251134, 2527711, 33985852, 611846939, 14864650923, 488222721991, 21712049275197, 1308300679611468, 106897965189674290, 11852113048215107821, 1784730721403509209214, 365323537513403184463272

Offset: 1

Eric W. Weisstein, Oct 06 2002

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..50
Keith M. Briggs, Combinatorial Graph Theory
Eric Weisstein's World of Mathematics, n-Chromatic Graph

Column k=2 of A084268.

Cf. A076279, A076280, A076281, A076282, A115597.

A005142 = Import["https://oeis.org/A005142/b005142.txt", "Table"][[All, 2]];
etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n - j], {j, 1, n}]/n]; b];
a = etr[A005142[[# + 1]]&][#] - 1&;
Array[a, 23] (* Jean-François Alcover, Sep 03 2019 *)

a(n) = A033995(n)-1.

More terms from Vladeta Jovovic, Jul 31 2003

Terms a(21) and beyond from Andrew Howroyd, Sep 05 2018

A076315 Number of 3-colorable (i.e., chromatic number <= 3) simple graphs on n nodes.

Original entry on oeis.org

1, 2, 4, 10, 29, 119, 667, 6024, 88500, 2109828, 78347534, 4383817811, 362181166439

Offset: 1

Eric W. Weisstein, Oct 06 2002

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

Cf. A033995, A076279, A076316, A076317, A076318, A076319, A076320, A076321.

a(n) = A033995(n) + A076279(n). - Andrew Howroyd, Dec 02 2018

a(10)-a(11) from Andrew Howroyd, Dec 02 2018
a(12) from Brendan McKay, Jan 19 2020
a(13) from Brendan McKay, Nov 08 2022

A076316 Number of 4-colorable (i.e., chromatic number <= 4) simple graphs on n nodes.

Original entry on oeis.org

1, 2, 4, 11, 33, 150, 985, 11390, 243791, 9965456, 753402410, 101344230844

Offset: 1

Eric W. Weisstein, Oct 06 2002

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

Cf. A033995, A076280, A076315, A076317, A076318, A076319, A076320, A076321.

a(n) = A076315(n) + A076280(n). - Andrew Howroyd, Dec 02 2018

a(10)-a(11) from Andrew Howroyd, Dec 02 2018

a(12) from Sean A. Irvine, Apr 13 2025

A076317 Number of 5-colorable (i.e., chromatic number <= 5) simple graphs on n nodes.

Original entry on oeis.org

1, 2, 4, 11, 34, 155, 1037, 12257, 272513, 11885351, 1003932892

Offset: 1

Eric W. Weisstein, Oct 06 2002

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

Cf. A033995, A076281, A076315, A076316, A076318, A076319, A076320, A076321.

Cf. A215620 (numbers of apex graphs on n vertices).

a(n) = A076316(n) + A076281(n). - Andrew Howroyd, Dec 02 2018

a(10)-a(11) from Andrew Howroyd, Dec 02 2018

cp's OEIS Frontend

A123408 Duplicate of A033995.

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A047863 Number of labeled graphs with 2-colored nodes where black nodes are only connected to white nodes and vice versa.

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A005142 Number of connected bipartite graphs with n nodes.

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A006785 Number of triangle-free graphs on n vertices.

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A047864 Number of labeled bipartite graphs with n nodes.

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A084268 Triangle read by rows: T(n,k) is the number of simple graphs on n unlabeled nodes having chromatic number k, 1 <= k <= n.

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Sage

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A076278 Number of 2-chromatic (i.e., chromatic number equals 2) simple graphs on n nodes.

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Mathematica

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A076315 Number of 3-colorable (i.e., chromatic number <= 3) simple graphs on n nodes.

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