A058843 -id:A058843 - cp's OEIS frontend

A003024 Number of acyclic digraphs (or DAGs) with n labeled nodes.

1, 1, 3, 25, 543, 29281, 3781503, 1138779265, 783702329343, 1213442454842881, 4175098976430598143, 31603459396418917607425, 521939651343829405020504063, 18676600744432035186664816926721, 1439428141044398334941790719839535103, 237725265553410354992180218286376719253505

Offset: 0

N. J. A. Sloane

Also the number of n X n real (0,1)-matrices with all eigenvalues positive. - Conjectured by Eric W. Weisstein, Jul 10 2003 and proved by McKay et al. 2003, 2004
Also the number of n X n real (0,1)-matrices with permanent equal to 1, up to permutation of rows/columns, cf. A089482. - Vladeta Jovovic, Oct 28 2009
Also the number of nilpotent elements in the semigroup of binary relations on [n]. - Geoffrey Critzer, May 26 2022
From Gus Wiseman, Jan 01 2024: (Start)
Also the number of sets of n nonempty subsets of {1..n} such that there is a unique way to choose a different element from each. For example, non-isomorphic representatives of the a(3) = 25 set-systems are:
  {{1},{2},{3}}
  {{1},{2},{1,3}}
  {{1},{2},{1,2,3}}
  {{1},{1,2},{1,3}}
  {{1},{1,2},{2,3}}
  {{1},{1,2},{1,2,3}}
These set-systems have ranks A367908, subset of A367906, for multisets A368101.
The version for no ways is A368600, any length A367903, ranks A367907.
The version for at least one way is A368601, any length A367902.
(End)

			For n = 2 the three (0,1)-matrices are {{{1, 0}, {0, 1}}, {{1, 0}, {1, 1}}, {{1, 1}, {0, 1}}}.

Archer, K., Gessel, I. M., Graves, C., & Liang, X. (2020). Counting acyclic and strong digraphs by descents. Discrete Mathematics, 343(11), 112041.
S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 310.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 19, Eq. (1.6.1).
R. W. Robinson, Counting labeled acyclic digraphs, pp. 239-273 of F. Harary, editor, New Directions in the Theory of Graphs. Academic Press, NY, 1973.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P Stanley, Enumerative Combinatorics I, 2nd. ed., p. 322.

Alois P. Heinz, Table of n, a(n) for n = 0..77 (first 41 terms from T. D. Noe)
T. E. Allen, J. Goldsmith, and N. Mattei, Counting, Ranking, and Randomly Generating CP-nets, 2014.
Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions, thesis, 2002.
Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions, thesis, 2002 [Local copy, with permission]
Eunice Y.-J. Chen, A. Choi, and A. Darwiche, On Pruning with the MDL Score, JMLR: Workshop and Conference Proceedings vol 52, 98-109, 2016.
S. Engstrom, Random acyclic orientations of graphs, Master's thesis written at the department of Mathematics at the Royal Institute of Technology (KTH) in Stockholm, Jan. 2013.
Zehao Jin, Mario Pasquato, Benjamin L. Davis, Tristan Deleu, Yu Luo, Changhyun Cho, Pablo Lemos, Laurence Perreault-Levasseur, Yoshua Bengio, Xi Kang, Andrea Valerio Maccio, and Yashar Hezaveh, A Data-driven Discovery of the Causal Connection between Galaxy and Black Hole Evolution, arXiv:2410.00965 [astro-ph.GA], 2024. See p. 33.
Zehao Jin, Mario Pasquato, Benjamin L. Davis, Andrea V. Macciò, and Yashar Hezaveh, Beyond Causal Discovery for Astronomy: Learning Meaningful Representations with Independent Component Analysis, arXiv:2410.14775 [astro-ph.GA], 2024. See pp. 2, 4, 10.
Zehao Jin, Mario Pasquato, Benjamin L. Davis, Tristan Deleu, Yu Luo, Changhyun Cho, Pablo Lemos, Laurence Perreault-Levasseur, Yoshua Bengio, and Xi Kang, Causal Discovery in Astrophysics: Unraveling Supermassive Black Hole and Galaxy Coevolution, Astrophys. J. (2025) Vol. 979, 212. See 4.1.
P. L. Krapivsky, Hiring Strategies, arXiv:2412.10490 [cs.GT], 2024. See p. 12.
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. 33.
Jack Kuipers and Giusi Moffa, Uniform generation of random acyclic digraphs, arXiv preprint arXiv:1202.6590 [stat.CO], 2012. - _N. J. A. Sloane_, Sep 14 2012
Laphou Lao, Zecheng Li, Songlin Hou, Bin Xiao, Songtao Guo, and Yuanyuan Yang, A Survey of IoT Applications in Blockchain Systems: Architecture, Consensus and Traffic Modeling, ACM Computing Surveys (CSUR, 2020) Vol. 53, No. 1, Article No. 18.
Tobias Ellegaard Larsen, Claus Thorn Ekstrøm, and Anne Helby Petersen, Score-Based Causal Discovery with Temporal Background Information, arXiv:2502.06232 [stat.ME], 2025. See p. 9.
Daniel Mallia, Towards an Unsupervised Bayesian Network Pipeline for Explainable Prediction, Decision Making and Discovery, Master's Thesis, CUNY Hunter College (2023).
B. D. McKay, F. E. Oggier, G. F. Royle, N. J. A. Sloane, I. M. Wanless and H. S. Wilf, Acyclic digraphs and eigenvalues of (0,1)-matrices, J. Integer Sequences, 7 (2004), #04.3.3.
B. D. McKay, F. E. Oggier, G. F. Royle, N. J. A. Sloane, I. M. Wanless and H. S. Wilf, Acyclic digraphs and eigenvalues of (0,1)-matrices, arXiv:math/0310423 [math.CO], Oct 28 2003.
A. Motzek and R. Möller, Exploiting Innocuousness in Bayesian Networks, Preprint 2015.
Yisu Peng, Y. Jiang, and P. Radivojac, Enumerating consistent subgraphs of directed acyclic graphs: an insight into biomedical ontologies, arXiv preprint arXiv:1712.09679 [cs.DS], 2017.
J. Peters, J. Mooij, D. Janzing, and B. Schölkopf, Causal Discovery with Continuous Additive Noise Models, arXiv preprint arXiv:1309.6779 [stat.ML], 2013.
R. W. Robinson, Enumeration of acyclic digraphs, Manuscript. (Annotated scanned copy)
V. I. Rodionov, On the number of labeled acyclic digraphs, Disc. Math. 105 (1-3) (1992) 319-321
I. Shpitser, T. S. Richardson, J. M. Robins and R. Evans, Parameter and Structure Learning in Nested Markov Models, arXiv preprint arXiv:1207.5058 [stat.ML], 2012.
I. Shpitser, R. J. Evans, T. S. Richardson, and J. M. Robins, Introduction to nested Markov models, Behaviormetrika, Behaviormetrika Vol. 41, No. 1, 2014, 3-39.
R. P. Stanley, Acyclic orientation of graphs, Discrete Math. 5 (1973), 171-178. North Holland Publishing Company.
Christian Toth, Christian Knoll, Franz Pernkopf, and Robert Peharz, Rao-Blackwellising Bayesian Causal Inference, arXiv:2402.14781 [cs.LG], 2024.
Sumanth Varambally, Yi-An Ma, and Rose Yu, Discovering Mixtures of Structural Causal Models from Time Series Data, arXiv:2310.06312 [cs.LG], 2023.
S. Wagner, Asymptotic enumeration of extensional acyclic digraphs, in Proceedings of the SIAM Meeting on Analytic Algorithmics and Combinatorics (ANALCO12).
Daniel Waxman, Kurt Butler, and Petar M. Djuric, Dagma-DCE: Interpretable, Non-Parametric Differentiable Causal Discovery, arXiv:2401.02930 [cs.LG], 2024.
Eric Weisstein's World of Mathematics, (0,1)-Matrix
Eric Weisstein's World of Mathematics, Acyclic Digraph
Eric Weisstein's World of Mathematics, Positive Eigenvalued Matrix
Eric Weisstein's World of Mathematics, Weisstein's Conjecture
Jun Wu and Mathias Drton, Partial Homoscedasticity in Causal Discovery with Linear Models, arXiv:2308.08959 [math.ST], 2023.
Chris Ying, Enumerating Unique Computational Graphs via an Iterative Graph Invariant, arXiv:1902.06192 [cs.DM], 2019.
Index entries for sequences related to binary matrices

Cf. A086510, A081064 (refined by # arcs), A307049 (by # descents).
Cf. A055165, which counts nonsingular {0, 1} matrices and A085656, which counts positive definite {0, 1} matrices.
Cf. A188457, A135079, A137435 (acyclic 3-multidigraphs), A188490.
For a unique sink we have A003025.
The unlabeled version is A003087.
These are the reverse-alternating sums of rows of A046860.
The weakly connected case is A082402.
A reciprocal version is A334282.
Row sums of A361718.
Cf. A003465, A058877, A058891, A059201, A088957, A133686, A323818.

p:=evalf(solve(sum((-1)^n*x^n/(n!*2^(n*(n-1)/2)), n=0..infinity) = 0, x), 50); M:=evalf(sum((-1)^(n+1)*p^n/((n-1)!*2^(n*(n-1)/2)), n=1..infinity), 40); # program for evaluation of constants p and M in the asymptotic formula, Vaclav Kotesovec, Dec 09 2013

a[0] = a[1] = 1; a[n_] := a[n] = Sum[ -(-1)^k * Binomial[n, k] * 2^(k*(n-k)) * a[n-k], {k, 1, n}]; Table[a[n], {n, 0, 13}](* Jean-François Alcover, May 21 2012, after PARI *)
Table[2^(n*(n-1)/2)*n! * SeriesCoefficient[1/Sum[(-1)^k*x^k/k!/2^(k*(k-1)/2),{k,0,n}],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, May 19 2015 *)
Table[Length[Select[Subsets[Subsets[Range[n]],{n}],Length[Select[Tuples[#],UnsameQ@@#&]]==1&]],{n,0,5}] (* Gus Wiseman, Jan 01 2024 *)

a(n)=if(n<1,n==0,sum(k=1,n,-(-1)^k*binomial(n,k)*2^(k*(n-k))*a(n-k)))

{a(n)=polcoeff(1-sum(k=0, n-1, a(k)*x^k/(1+2^k*x+x*O(x^n))^(k+1)), n)} \\ Paul D. Hanna, Oct 17 2009

a(0) = 1; for n > 0, a(n) = Sum_{k=1..n} (-1)^(k+1)*C(n, k)*2^(k*(n-k))*a(n-k).
1 = Sum_{n>=0} a(n)*exp(-2^n*x)*x^n/n!. - Vladeta Jovovic, Jun 05 2005
a(n) = Sum_{k=1..n} (-1)^(n-k)*A046860(n,k) = Sum_{k=1..n} (-1)^(n-k)*k!*A058843(n,k). - Vladeta Jovovic, Jun 20 2008
1 = Sum_{n=>0} a(n)*x^n/(1 + 2^n*x)^(n+1). - Paul D. Hanna, Oct 17 2009
1 = Sum_{n>=0} a(n)*C(n+m-1,n)*x^n/(1 + 2^n*x)^(n+m) for m>=1. - Paul D. Hanna, Apr 01 2011
log(1+x) = Sum_{n>=1} a(n)*(x^n/n)/(1 + 2^n*x)^n. - Paul D. Hanna, Apr 01 2011
Let E(x) = Sum_{n >= 0} x^n/(n!*2^C(n,2)). Then a generating function for this sequence is 1/E(-x) = Sum_{n >= 0} a(n)*x^n/(n!*2^C(n,2)) = 1 + x + 3*x^2/(2!*2) + 25*x^3/(3!*2^3) + 543*x^4/(4!*2^6) + ... (Stanley). Cf. A188457. - Peter Bala, Apr 01 2013
a(n) ~ n!*2^(n*(n-1)/2)/(M*p^n), where p = 1.488078545599710294656246... is the root of the equation Sum_{n>=0} (-1)^n*p^n/(n!*2^(n*(n-1)/2)) = 0, and M = Sum_{n>=1} (-1)^(n+1)*p^n/((n-1)!*2^(n*(n-1)/2)) = 0.57436237330931147691667... Both references to the article "Acyclic digraphs and eigenvalues of (0,1)-matrices" give the wrong value M=0.474! - Vaclav Kotesovec, Dec 09 2013 [Response from N. J. A. Sloane, Dec 11 2013: The value 0.474 has a typo, it should have been 0.574. The value was taken from Stanley's 1973 paper.]
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 2*x^2 + 10*x^3 + 146*x^4 + 6010*x^5 + ... appears to have integer coefficients (cf. A188490). - Peter Bala, Jan 14 2016

A240936 Number of ways to partition the (vertex) set {1,2,...,n} into any number of classes and then select some unordered pairs (edges) such that a and b are in distinct classes of the partition.

Original entry on oeis.org

1, 1, 3, 21, 337, 11985, 930241, 155643329, 55638770689, 42200814258433, 67536939792143361, 227017234854393949185, 1596674435594864988020737, 23421099407847007850007154689, 714530983411175509576743561314305, 45227689798343820164634911814524846081

Offset: 0

Geoffrey Critzer, Aug 03 2014

nonn

The elements of a class are allowed to be used multiple times to form the unordered pairs.
Equivalently, a(n) is the sum of the number of k-colored graphs on n labeled nodes taken over k colors, 1<=k<=n, where labeled graphs using k colors that differ only by a permutation of the k colors are considered to be the same.
Also the number of ways to choose a stable partition of a simple graph on n vertices. A stable partition of a graph is a set partition of the vertices where no edge has both ends in the same block. - Gus Wiseman, Nov 24 2018

			a(2)=3 because the empty graph with 2 nodes is counted twice (once for each partition of 2) and the complete graph is counted once. 2+1=3.

Alois P. Heinz, Table of n, a(n) for n = 0..75

Cf. A000569, A001187, A006125, A058843, A277203, A321979.

b:= proc(n, k) b(n, k):= `if`(k=1, 1, add(binomial(n, i)*
      2^(i*(n-i))*b(i, k-1)/k, i=1..n-1))
    end:
a:= n-> `if`(n=0, 1, add(b(n, k), k=1..n)):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 04 2014

nn=15;e[x_]:=Sum[x^n/(n!*2^Binomial[n,2]),{n,0,nn}];Table[n!2^Binomial[n,2],{n,0,nn}]CoefficientList[Series[Exp[(e[x]-1)],{x,0,nn}],x]

seq(n)={Vec(serconvol(sum(j=0, n, x^j*j!*2^binomial(j,2)) + O(x*x^n), exp(sum(j=1, n, x^j/(j!*2^binomial(j, 2))) + O(x*x^n))))} \\ Andrew Howroyd, Dec 01 2018

a(n) = n! * 2^C(n,2) * [x^n] exp(E(x)-1) where E(x) is Sum_{n>=0} x^n/(n!*2^C(n,2)).

a(n) = Sum_{k=1..n} A058843(n,k) for n>0.

A322280 Array read by antidiagonals: T(n,k) is the number of graphs on n labeled nodes, each node being colored with one of k colors, where no edge connects two nodes of the same color.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 6, 1, 0, 1, 4, 15, 26, 1, 0, 1, 5, 28, 123, 162, 1, 0, 1, 6, 45, 340, 1635, 1442, 1, 0, 1, 7, 66, 725, 7108, 35043, 18306, 1, 0, 1, 8, 91, 1326, 20805, 254404, 1206915, 330626, 1, 0, 1, 9, 120, 2191, 48486, 1058885, 15531268, 66622083, 8488962, 1, 0

Offset: 0

Andrew Howroyd, Dec 01 2018

Not all colors need to be used.

			Array begins:
===============================================================
n\k| 0 1      2        3          4           5           6
---+-----------------------------------------------------------
0  | 1 1      1        1          1           1           1 ...
1  | 0 1      2        3          4           5           6 ...
2  | 0 1      6       15         28          45          66 ...
3  | 0 1     26      123        340         725        1326 ...
4  | 0 1    162     1635       7108       20805       48486 ...
5  | 0 1   1442    35043     254404     1058885     3216486 ...
6  | 0 1  18306  1206915   15531268    95261445   386056326 ...
7  | 0 1 330626 66622083 1613235460 15110296325 83645197446 ...
...

Andrew Howroyd, Table of n, a(n) for n = 0..1274
R. C. Read, The number of k-colored graphs on labelled nodes, Canad. J. Math., 12 (1960), 410-414.
R. C. Read and E. M. Wright, Colored graphs: A correction and extension, Canad. J. Math. 22 1970 594-596.

Columns k=0..4 are A000007, A000012, A047863, A191371, A223887.
Main diagonal gives A372920.
Cf. A058843, A058875, A322278, A322279.

nmax = 10;
T[n_, k_] := n!*2^Binomial[n, 2]*SeriesCoefficient[Sum[ x^i/(i!* 2^Binomial[i, 2]), {i, 0, nmax}]^k, {x, 0, n}];
Table[T[n - k, k], {n, 0, nmax}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Sep 23 2019 *)

M(n)={
  my(p=sum(j=0, n, x^j/(j!*2^binomial(j, 2))) + O(x*x^n));
  my(q=sum(j=0, n, x^j*j!*2^binomial(j, 2)) + O(x*x^n));
  matconcat([1, Mat(vector(n, k, Col(serconvol(q, p^k))))]);
}
my(T=M(7)); for(n=1, #T, print(T[n,]))

T(n,k) = n!*2^binomial(n,2) * [x^n](Sum_{i>=0} x^i/(i!*2^binomial(i,2)))^k.

T(n,k) = Sum_{j=0..k} binomial(k,j)*j!*A058843(n,j).

A000683 Number of colorings of labeled graphs on n nodes using exactly 2 colors, divided by 4.

Original entry on oeis.org

0, 1, 6, 40, 360, 4576, 82656, 2122240, 77366400, 4002843136, 293717546496, 30558458490880, 4505780560619520, 941417163728674816, 278628902101315608576, 116805328001281573519360, 69340603828363322892779520, 58287619305053298399714082816, 69366390252412220606233109200896

Offset: 1

N. J. A. Sloane

A coloring of a simple graph is a choice of color for each graph vertex such that no two vertices sharing the same edge have the same color. A213441 counts those colorings of labeled graphs on n vertices that use exactly two colors. This sequence is 1/4 of A213441 (1/4 of column 2 of Table 1 in Read). - Peter Bala, Apr 11 2013

A047863 counts colorings of labeled graphs on n vertices that use two or fewer colors. - Peter Bala, Apr 11 2013

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 18, table 1.5.1, column 2 (divided by 2).
R. C. Read, personal communication.
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).

T. D. Noe, Table of n, a(n) for n=1..50
R. C. Read, The number of k-colored graphs on labelled nodes, Canad. J. Math., 12 (1960), 410-414.

a(n)=(A047863(n)-2)/4.
A diagonal of A058843.
One quarter of A213441.

maxn = 16; t[, 1] = 1; t[n, k_] := t[n, k] = Sum[Binomial[n, j]*2^(j*(n - j))*t[j, k - 1]/k, {j, 1, n - 1}]; a[n_] := t[n, 2]/2; Table[a[n], {n, 1, maxn}] (* Jean-François Alcover, Sep 21 2011 *)

Reference gives generating function.

a(n) ~ c * 2^(n^2/4+n-3/2)/sqrt(Pi*n), where c = Sum_{k = -infinity..infinity} 2^(-k^2) = 2.128936827211877... if n is even and c = Sum_{k = -infinity..infinity} 2^(-(k+1/2)^2) = 2.12893125051302... if n is odd. - Vaclav Kotesovec, Jun 24 2013

More terms from Vladeta Jovovic, Feb 02 2000

A322279 Array read by antidiagonals: T(n,k) is the number of connected graphs on n labeled nodes, each node being colored with one of k colors, where no edge connects two nodes of the same color.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 2, 0, 0, 1, 4, 6, 6, 0, 0, 1, 5, 12, 42, 38, 0, 0, 1, 6, 20, 132, 618, 390, 0, 0, 1, 7, 30, 300, 3156, 15990, 6062, 0, 0, 1, 8, 42, 570, 9980, 136980, 668526, 134526, 0, 0, 1, 9, 56, 966, 24330, 616260, 10015092, 43558242, 4172198, 0, 0

Offset: 0

Andrew Howroyd, Dec 01 2018

Not all colors need to be used.

			Array begins:
===============================================================
n\k| 0 1      2        3          4           5           6
---+-----------------------------------------------------------
0  | 1 1      1        1          1           1           1 ...
1  | 0 1      2        3          4           5           6 ...
2  | 0 0      2        6         12          20          30 ...
3  | 0 0      6       42        132         300         570 ...
4  | 0 0     38      618       3156        9980       24330 ...
5  | 0 0    390    15990     136980      616260     1956810 ...
6  | 0 0   6062   668526   10015092    65814020   277164210 ...
7  | 0 0 134526 43558242 1199364852 11878194300 67774951650 ...
...

Andrew Howroyd, Table of n, a(n) for n = 0..1274
R. C. Read, E. M. Wright, Colored graphs: A correction and extension, Canad. J. Math. 22 1970 594-596.

Columns k=2..5 are A002027, A002028, A002029, A002030.

Cf. A058843, A058875, A322278, A322280.

M(n)={
  my(p=sum(j=0, n, x^j/(j!*2^binomial(j, 2))) + O(x*x^n));
  my(q=sum(j=0, n, x^j*2^binomial(j, 2)) + O(x*x^n));
  my(W=Mat(vector(n, k, Col(serlaplace(1 + log(serconvol(q, p^k)))))));
  matconcat([1, W]);
}
my(T=M(7)); for(n=1, #T, print(T[n,]))

k-th column is the logarithmic transform of the k-th column of A322280.

E.g.f of k-th column: 1 + log(Sum_{n>=0} A322280(n,k)*x^n/n!).

A213441 Number of 2-colored graphs on n labeled nodes.

Original entry on oeis.org

0, 4, 24, 160, 1440, 18304, 330624, 8488960, 309465600, 16011372544, 1174870185984, 122233833963520, 18023122242478080, 3765668654914699264, 1114515608405262434304, 467221312005126294077440, 277362415313453291571118080, 233150477220213193598856331264, 277465561009648882424932436803584, 467466753447825987214906927108587520

Offset: 1

N. J. A. Sloane, Jun 11 2012

From Peter Bala, Apr 10 2013: (Start)
A coloring of a simple graph is a choice of color for each graph vertex such that no two vertices sharing the same edge have the same color. This sequence counts only those colorings of labeled graphs on n vertices that use exactly two colors.
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 Read has shown that (E(x) - 1)^k is a generating function for counting labeled graphs colored using precisely k colors. This is the case k = 2. For other cases see A213442 (k = 3) and A224068 (k = 4).
A coloring of a graph G that uses k or fewer colors is called a k-coloring of G. The graph G is k-colored if a k-coloring of G exists.
Then E(x)^k is a generating function for the enumeration of labeled k-colored graphs on n vertices (see Stanley). For cases see A047863 (k = 2), A191371 (k = 3) and A223887 (k = 4). (End)

			a(2) = 4: Denote the vertex coloring by o and *. The 4 labeled graphs on 2 vertices that can be colored using exactly two colors are
....1....2............1....2
....o....*............*----o
...........................
....1....2............1....2
....*....o............o----*
- _Peter Bala_, Apr 10 2013

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

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. Volume 306, Issues 10-11, 28 May 2006, Pages 905-909.
Eric Weisstein's World of Mathematics, k-Colorable Graph
Wikipedia, Graph coloring

Cf. A047863, A058843, A191371, A213442, A223887, A224068.

F2:=n->add(binomial(n,r)*2^(r*(n-r)), r=1..n-1);
[seq(F2(n),n=1..20)];

nn=20;a[x_]:=Sum[x^n/(n!*(2^(n^2/2))),{n,0,nn}];Drop[Table[n!*(2^(n^2/2)),{n,0,nn}]CoefficientList[Series[(a[x]-1)^2,{x,0,nn}],x],1] (* Geoffrey Critzer, Aug 05 2014 *)

a(n) = sum(k=1,n-1, binomial(n,k)*2^(k*(n-k)) ); /* Joerg Arndt, Apr 10 2013 */

From Peter Bala, Apr 10 2013: (Start)
a(n) = Sum_{k = 1..n-1} binomial(n,k)*2^(k*(n-k)). a(n) = A047863(n) - 2.
Let E(x) = Sum_{n >= 0} x^n/(n!*2^C(n,2)) = 1 + x + x^2/(2!*2) + x^3/(3!*2^3) + x^4/(4!*2^6) + .... Then a generating function is (E(x) - 1)^2 = 4*x^2/(2!*2) + 24*x^3/(3!*2^3) + 160*x^4/(4!*2^6) + ....
Sequence is 1/2*(column 2) from A058843. (End)
E.g.f.: Sum_{n>=1}(exp(2^n*x)-1)*x^n/n!. - Geoffrey Critzer, Aug 11 2014

A322278 Triangle read by rows: T(n,k) is the number of k-colored connected graphs on n labeled nodes up to permutation of the colors.

Original entry on oeis.org

1, 0, 1, 0, 3, 4, 0, 19, 84, 38, 0, 195, 2470, 3140, 728, 0, 3031, 108390, 307390, 186360, 26704, 0, 67263, 7192444, 42747460, 52630060, 18926544, 1866256, 0, 2086099, 726782784, 9030799218, 20784069600, 14401134944, 3463311488, 251548592

Offset: 1

Andrew Howroyd, Dec 01 2018

Equivalently, the number of ways to choose a stable partition of a simple connected graph on n labeled nodes with k parts. See A322064 for the definition of stable partition.

			Triangle begins:
  1;
  0,     1;
  0,     3,       4;
  0,    19,      84,       38;
  0,   195,    2470,     3140,      728;
  0,  3031,  108390,   307390,   186360,    26704;
  0, 67263, 7192444, 42747460, 52630060, 18926544, 1866256;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Row sums are A322064.
Columns k=2..4 are A001832(for n > 1), A322330, A322331.
Right diagonal is A001187.
Cf. A058843, A058875, A322279, A322280.

M(n, K=n)={
  my(p=sum(j=0, n, x^j/(j!*2^binomial(j, 2))) + O(x*x^n));
  my(q=sum(j=0, n, x^j*2^binomial(j, 2)) + O(x*x^n));
  my(W=vector(K, k, Col(serlaplace(log(serconvol(q, p^k))))));
  Mat(vector(K, k, sum(i=1, k, (-1)^(k-i)*binomial(k,i)*W[i])/k!));
}
my(T=M(7)); for(n=1, #T, print(T[n, 1..n]))

T(n,k) = (1/k!)*Sum_{j=0..k} (-1)^(k-j)*binomial(k,j)*A322279(n,j).

A058875 Triangle T(n,k) = C_n(k)/2^(k*(k-1)/2) where C_n(k) = number of k-colored labeled graphs with n nodes (n >= 1, 1 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 40, 24, 1, 1, 360, 640, 80, 1, 1, 4576, 24000, 7040, 240, 1, 1, 82656, 1367296, 878080, 62720, 672, 1, 1, 2122240, 122056704, 169967616, 23224320, 487424, 1792, 1, 1, 77366400, 17282252800, 53247344640, 13440516096

Offset: 1

N. J. A. Sloane, Jan 07 2001

From Peter Bala, Apr 12 2013: (Start)
A coloring of a simple graph G is a choice of color for each graph vertex such that no two vertices sharing the same edge have the same color.
Let E(x) = Sum_{n >= 0} x^n/(n!*2^C(n,2)) = 1 + x + x^2/(2*2!) + x^3/(2^3*3!) + .... Read has shown that (E(x) - 1)^k is a generating function for labeled graphs on n nodes that can be colored using exactly k colors. Cases include A213441 (k = 2), A213442 (k = 3) and A224068 (k = 4).
If colorings of a graph using k colors are counted as the same if they differ only by a permutation of the colors then a generating function is 1/k!*(E(x) - 1)^k , which is a generating function for the k-th column of A058843. Removing a further factor of 2^C(k,2) gives 1/(k!*2^C(k,2))*(E(x) - 1)^k as a generating function for the k-th column of this triangle. (End)

			Triangle begins:
  1;
  1,     1;
  1,     6,       1;
  1,    40,      24,      1;
  1,   360,     640,     80,     1;
  1,  4576,   24000,   7040,   240,   1;
  1, 82656, 1367296, 878080, 62720, 672, 1;
  ...

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 18, Table 1.5.1.

Andrew Howroyd, Table of n, a(n) for n = 1..1275
Steven R. Finch, Bipartite, k-colorable and k-colored graphs
Steven R. Finch, Bipartite, k-colorable and k-colored graphs, June 5, 2003. [Cached copy, with permission of the author]
R. C. Read, The number of k-colored graphs on labelled nodes, Canad. J. Math., 12 (1960), 410-414.
Eric Weisstein's World of Mathematics, k-Colorable Graph

Apart from scaling, same as A058843.
Columns give A058872 and A000683, A058873 and A006201, A058874 and A006202, also A006218.
Cf. A213441, A213442, A224068.

maxn=8; t[,1]=1; t[n,k_]:=t[n,k]=Sum[Binomial[n,j]*2^(j*(n-j))*t[j,k-1]/k,{j,1,n-1}]; Flatten[Table[t[n,k]/2^Binomial[k,2], {n,1,maxn},{k,1,n}]]  (* Geoffrey Critzer, Oct 06 2012, after code from Jean-François Alcover in A058843 *)

b(n)={n!*2^binomial(n,2)}
T(n,k)={b(n)*polcoef((sum(j=1, n, x^j/b(j)) + O(x*x^n))^k, n)/b(k)} \\ Andrew Howroyd, Nov 30 2018

C_n(k) = Sum_{i=1..n-1} binomial(n, i)*2^(i*(n-i))*C_i(k-1)/k.
From Peter Bala, Apr 12 2013: (Start)
Recurrence equation: T(n,k) = 1/2^(k-1)*Sum_{i = 1..n-1} binomial(n-1,i)*2^(i*(n-i))*T(i,k-1).
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 for this triangle is E(x*(E(z) - 1)) = 1 + x*z + (x + x^2 )*z^2/(2!*2) + (x + 6*x^2 + x^3)*z^3/(3!*2^3) + (x + 40*x^2 + 24*x^3 + x^4)*z^4/(4!*2^6) + .... Cf. A008277 with e.g.f. exp(x*(exp(z) - 1)).
The row polynomials R(n,x) satisfy the recurrence equation R(n,x) = x*sum {k = 0..n-1} binomial(n-1,k)*2^(k*(n-k))*R(k,x/2) with R(0,x) = 1. The row polynomials appear to have only real zeros.
Column 2 = 1/(2!*2)*A213441; column 3 = 1/(3!*2^3)*A213442; column 4 = 1/(4!*2^6)*A224068. (End)
T(n,k) = A058843(n,k)/2^binomial(k,2). - Andrew Howroyd, Nov 30 2018

cp's OEIS Frontend

A003024 Number of acyclic digraphs (or DAGs) with n labeled nodes.

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A240936 Number of ways to partition the (vertex) set {1,2,...,n} into any number of classes and then select some unordered pairs (edges) such that a and b are in distinct classes of the partition.

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A322280 Array read by antidiagonals: T(n,k) is the number of graphs on n labeled nodes, each node being colored with one of k colors, where no edge connects two nodes of the same color.

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A000683 Number of colorings of labeled graphs on n nodes using exactly 2 colors, divided by 4.

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A322279 Array read by antidiagonals: T(n,k) is the number of connected graphs on n labeled nodes, each node being colored with one of k colors, where no edge connects two nodes of the same color.

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A213441 Number of 2-colored graphs on n labeled nodes.

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A322278 Triangle read by rows: T(n,k) is the number of k-colored connected graphs on n labeled nodes up to permutation of the colors.

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