A047863 -id:A047863 - cp's OEIS frontend

A001831 Number of labeled graded partially ordered sets with n elements of height at most 1.

1, 1, 3, 13, 87, 841, 11643, 227893, 6285807, 243593041, 13262556723, 1014466283293, 109128015915207, 16521353903210521, 3524056001906654763, 1059868947134489801413, 449831067019305308555487, 269568708630308018001547681, 228228540531327778410439620963

Offset: 0

N. J. A. Sloane

Labeled posets where for all a,b,c in the set, do not have a

Number of labeled digraphs with n vertices with no directed path of length 2. Number of n X n {0,1} matrices A such that A^2 = 0. - Michael Somos, Jul 28 2013

Number of relations on n labeled nodes that are simultaneously transitive and antitransitive. - Peter Kagey, Feb 14 2021

			1 + x + 3*x^2 + 13*x^3 + 87*x^4 + 841*x^5 + 11643*x^6 + 227893*x^7 + ...

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 = 0..50
D. A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19. [Annotated scanned copy]
D. A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19.
A. Motzek and R. Möller, Exploiting Innocuousness in Bayesian Networks, Preprint, Australasian Joint Conference on Artificial Intelligence AI 2015: Advances in Artificial Intelligence, pp. 411-423.
Zvi Rosen and Yan X. Zhang, Convex Neural Codes in Dimension 1, arXiv:1702.06907 [math.CO], 2017. Mentions this sequence.
Index entries for sequences related to posets

Cf. A002031, A047863, A052332, A001833, A048194.
Row sums of A052296.
Cf. variants: A135753, A135754.

A001831 := proc(n)
    add(binomial(n,k)*(2^k-1)^(n-k),k=0..n) ;
end proc:
seq(A001831(n),n=0..10) ; # R. J. Mathar, Mar 08 2021

Join[{1}, Table[Sum[Binomial[n,k](2^k-1)^(n-k),{k,n}],{n,20}]] (* Harvey P. Dale, Jan 05 2012 *)

{a(n)=n!*polcoeff(sum(k=0,n,exp((2^k-1)*x)*x^k/k!),n)} \\ Paul D. Hanna, Nov 27 2007

{a(n)=polcoeff(sum(k=0, n, x^k/(1-(2^k-1)*x +x*O(x^n))^(k+1)), n)} \\ Paul D. Hanna, Sep 15 2009

a(n) = Sum((-1)^k*C(n, k)*A047863(k), k=0..n).
a(n) = Sum_{k=0..n} binomial(n, k)*(2^k-1)^(n-k). - Vladeta Jovovic, Apr 04 2003
E.g.f.: Sum_{n>=0} exp((2^n-1)*x) * x^n/n!. - Paul D. Hanna, Nov 27 2007 [correction made by Paul D. Hanna, Mar 08 2021]
O.g.f.: Sum_{n>=0} x^n/(1 - (2^n - 1)*x)^(n+1) = Sum_{n>=0} a(n)*x^n. - Paul D. Hanna, Sep 15 2009
a(n) ~ c * 2^(n^2/4 + n + 1/2) / sqrt(Pi*n), where c = JacobiTheta3(0,1/2) = EllipticTheta[3, 0, 1/2] = 2.1289368272118771586694585485449... if n is even, and c = JacobiTheta2(0,1/2) = EllipticTheta[2, 0, 1/2] = 2.1289312505130275585916134025753... if n is odd. - Vaclav Kotesovec, Mar 10 2014

More terms, formula and comments from Christian G. Bower, Dec 15 1999
Last 4 terms corrected by Vladeta Jovovic, Apr 04 2003
Comments corrected by Joel B. Lewis, Mar 28 2011

A001832 Number of labeled connected bipartite graphs on n nodes.

Original entry on oeis.org

1, 1, 3, 19, 195, 3031, 67263, 2086099, 89224635, 5254054111, 426609529863, 47982981969979, 7507894696005795, 1641072554263066471, 502596525992239961103, 216218525837808950623459, 130887167385831881114006475, 111653218763166828863141636911

Offset: 1

N. J. A. Sloane

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 406.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
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
F. Harary and R. W. Robinson, Labeled bipartite blocks, Canad. J. Math., 31 (1979), 60-68.
F. Harary and R. W. Robinson, Labeled bipartite blocks, Canad. J. Math., 31 (1979), 60-68. (Annotated scanned copy)
D. A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19.
D. A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19. [Annotated scanned copy]
A. Nymeyer and R. W. Robinson, Tabulation of the Numbers of Labeled Bipartite Blocks and Related Classes of Bicolored Graphs, 1982 [Annotated scanned copy of unpublished MS and letter from R.W.R.]
Eric Weisstein's World of Mathematics, n-Colorable Graph
Eric Weisstein's World of Mathematics, n-Chromatic Graph

Row sums of A228861.
The unlabeled version is A005142.
Cf. A002031, A047863, A047864.

mx = 17; s = Sum[ Binomial[n, k] 2^(k (n - k)) x^n/n!, {n, 0, mx}, {k, 0, n}] ; Range[0, mx]! CoefficientList[ Series[ Log[s]/2, {x, 0, mx}], x] (* Geoffrey Critzer, May 10 2011 *)

seq(n)=Vec(serlaplace(log(sum(k=0, n, exp(2^k*x + O(x*x^n))*x^k/k!))/2)) \\ Andrew Howroyd, Sep 26 2018

E.g.f.: log(A(x))/2 where A(x) is e.g.f. of A047863.

a(n) = A002031(n)/2, for n > 1. - Geoffrey Critzer, May 10 2011

More terms from Vladeta Jovovic, Apr 12 2003

A002031 Number of labeled connected digraphs on n nodes where every node has indegree 0 or outdegree 0 and no isolated nodes.

Original entry on oeis.org

2, 6, 38, 390, 6062, 134526, 4172198, 178449270, 10508108222, 853219059726, 95965963939958, 15015789392011590, 3282145108526132942, 1005193051984479922206, 432437051675617901246918, 261774334771663762228012950, 223306437526333657726283273822

Offset: 2

N. J. A. Sloane

Also number of labeled connected graphs with 2-colored nodes with no isolated nodes where black nodes are only connected to white nodes and vice versa.

In- or outdegree zero implies loops are not admitted. Multi-arcs are not admitted. - R. J. Mathar, Nov 18 2023

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).

Alois P. Heinz, Table of n, a(n) for n = 2..100
R. C. Read, E. M. Wright, Colored graphs: A correction and extension, Canad. J. Math. 22 1970 594-596.
N. J. A. Sloane, Transforms

Cf. A001831, A001832, A002032, A047863, A052332, A007776 (unlabeled case). Essentially the same as A002027.

logtr:= proc(p) local b; b:=proc(n) option remember; local k; if n=0 then 1 else p(n)- add(k *binomial(n,k) *p(n-k) *b(k), k=1..n-1)/n fi end end: digr:= n-> add(binomial(n,k) *(2^k-2)^(n-k), k=0..n): a:= logtr(digr): seq(a(n), n=2..25);  # Alois P. Heinz, Sep 14 2008

terms = 17; s = Log[Sum[Exp[(2^n - 2)*x]*(x^n/n!), {n, 0, terms+2}]] + O[x]^(terms+2); Drop[CoefficientList[s, x]*Range[0, terms+1]!, 2] (* Jean-François Alcover, Nov 08 2011, after Vladeta Jovovic, updated Jan 12 2018 *)

Logarithmic transform of A052332.
E.g.f.: log(Sum(exp((2^n-2)*x)*x^n/n!, n=0..infinity)). - Vladeta Jovovic, May 28 2004
a(n) = f(n,2) using functions defined in A002032. - Sean A. Irvine, May 29 2013

More terms, formula and new title from Christian G. Bower, Dec 15 1999

Corrected by Vladeta Jovovic, Apr 12 2003

A191371 Number of simple labeled graphs with (at most) 3-colored nodes such that no edge connects two nodes of the same color.

Original entry on oeis.org

1, 3, 15, 123, 1635, 35043, 1206915, 66622083, 5884188675, 830476531203, 187106645932035, 67241729173555203, 38521811621470420995, 35161184767296890265603, 51112793797110111859802115, 118291368253025368001553530883, 435713124846749574718274002747395, 2553666761436949125065383836043837443

Offset: 0

Geoffrey Critzer, Jun 01 2011

Cf. A213442, which counts colorings of labeled graphs on n vertices using exactly 3 colors. For 3-colorable labeled graphs on n vertices see A076315. - Peter Bala, Apr 12 2013

			a(2) = 15: There are two labeled 3-colorable graphs on 2 nodes, namely
A)  1    2   B)  1    2
    o    o       o----o
Using 3 colors there are 9 ways to color the graph of type A and 3*2 = 6 ways to color the graph of type B so that adjacent vertices do not share the same color. Thus there are in total 15 labeled 3-colored graphs on 2 vertices.

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, 1973, 16-18.

Andrew Howroyd, Table of n, a(n) for n = 0..50
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.
R. P. Stanley, Acyclic orientation of graphs, Discrete Math. 5 (1973), 171-178. North Holland Publishing Company.
Eric Weisstein's World of Mathematics, k-Colorable Graph

Column k=3 of A322280.

Cf. A047863. A000685, A076315, A223887.

f[{k_,r_,m_}]:= Binomial[m+r+k,k] Binomial[m+r,r] 2^ (k r + k m + r m); Table[Total[Map[f,Compositions[n,3]]],{n,0,20}]

seq(n)={my(p=(sum(j=0, n, x^j/(j!*2^binomial(j, 2))) + O(x*x^n))^3); Vecrev(sum(j=0, n, j!*2^binomial(j,2)*polcoef(p,j)*x^j))} \\ Andrew Howroyd, Dec 03 2018

from sympy import binomial
def a047863(n): return sum([binomial(n, k)*2**(k*(n - k)) for k in range(n + 1)])
def a(n): return sum([binomial(n, k)*2**(k*(n - k))*a047863(k) for k in range(n + 1)]) # Indranil Ghosh, Jun 03 2017

a(n) = Sum(C(n,k)*C(n-k,r)*2^(k*r+k*m+r*m)) where the sum is taken over all nonnegative solutions to k + r + m = n.
From Peter Bala, Apr 12 2013: (Start)
a(n) = Sum_{k = 0..n} binomial(n,k)*2^(k*(n-k))*A047863(k).
a(n) = 3*A000685(n) for n >= 1.
Let E(x) = Sum_{n >= 0} x^n/(n!*2^C(n,2)). Then a generating function for this sequence is E(x)^3 = sum {n >= 0} a(n)*x^n/(n!*2^C(n,2)) = 1 + 3*x + 15*x^2/(2!*2) + 123*x^3/(3!*2^3) + 1635*x^4/(4!*2^6) + ....
In general, for k = 1, 2, ..., E(x)^k is a generating function for labeled k-colored graphs (see Read). For other examples see A047863 (k = 2) and A223887 (k = 4). (End)

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 ).

A049312 Number of graphs with a distinguished bipartite block, by number of vertices.

Original entry on oeis.org

1, 2, 4, 8, 17, 38, 94, 258, 815, 3038, 13804, 78760, 580456, 5647602, 73645352, 1297920850, 31031370360, 1007551636038, 44432872400460, 2661065508648436, 216457998880015366, 23920728651724212120, 3593384834863975164882, 734240676501745813835934

Offset: 0

Peter J. Cameron

Calculate number of connected bipartite graphs + number of connected bipartite graphs with no duality automorphism, apply EULER transform.

Inverse Euler transform is A318870.

			a(2)=4: null graph with 0, 1 or 2 vertices in the distinguished block and complete graph with 1 vertex in distinguished block.

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

Alois P. Heinz, Table of n, a(n) for n = 0..40
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Karen L. Collins, Ann N. Trenk, Finding Balance: Split Graphs and Related Classes, arXiv:1706.03092 [math.CO], June 2017.
M. Guay-Paquet, A. H. Morales and E. Rowland, Structure and enumeration of (3+ 1)-free posets, arXiv preprint arXiv:1212.5356 [math.CO], 2012-2013. - From _N. J. A. Sloane_, Feb 01 2013
J. M. Troyka, Split graphs: combinatorial species and asymptotics, arXiv:1803.07248 [math.CO], 2018-2019.
J. M. Troyka, Split graphs: combinatorial species and asymptotics, Electron. J. Combin., 26 (2019), #P2.42.
E. M. Wright, The k-connectedness of bipartite graphs, J. Lond. Math. Soc. (2), 25 (1982), 7-12.

Row sums of A028657.

Cf. A048194, A318870.

b:= proc(n, i) option remember; `if`(n=0, {0}, `if`(i<1, {},
      {seq(map(p-> p+j*x^i, b(n-i*j, i-1) )[], j=0..n/i)}))
    end:
g:= proc(n, k) option remember; add(add(2^add(add(igcd(i, j)*
      coeff(s, x, i)* coeff(t, x, j), j=1..degree(t)),
      i=1..degree(s))/mul(i^coeff(s, x, i)*coeff(s, x, i)!,
      i=1..degree(s))/mul(i^coeff(t, x, i)*coeff(t, x, i)!,
      i=1..degree(t)), t=b(n+k$2)), s=b(n$2))
    end:
A:= (n, k)-> g(min(n, k), abs(n-k)):
a:= d-> add(A(n, d-n), n=0..d):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 01 2014

b[n_, i_] := b[n, i] = If[n == 0, {0}, If[i<1, {}, Flatten @ Table[ Map[ Function[ {p}, p+j*x^i], b[n-i*j, i-1]], {j, 0, n/i}]]];
g[n_, k_] := g[n, k] = Sum[ Sum[ 2^Sum[Sum[GCD[i, j]*Coefficient[s, x, i]*Coefficient[t, x, j], {j, 1, Exponent[t, x]}], {i, 1, Exponent[s, x]}]/Product[i^Coefficient[s, x, i]*Coefficient[s, x, i]!, {i, 1, Exponent[s, x]}]/Product[i^Coefficient[t, x, i]*Coefficient[t, x, i]!, {i, 1, Exponent[t, x]}], {t, b[n+k, n+k]}], {s, b[n, n]}];
A[n_, k_] := g[Min[n, k], Abs[n-k]];
a[d_] := Sum[A[n, d-n], {n, 0, d}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 25 2015, after Alois P. Heinz *)

a(n) ~ 1/n! A047863(n) = 1/n! Sum_{k=0..n} binomial(n,k) * 2^(k(n-k)) (see Wright; see also Thm. 3.7 of the Troyka link, which cites Wright). - Justin M. Troyka, Oct 29 2018

More terms from Vladeta Jovovic, Jun 17 2000

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

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cp's OEIS Frontend

A060224 Number of orbits of length n under the map whose periodic points are counted by A047863.

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A251684 G.f.: exp( Sum_{n>=1} A047863(n)*x^n/n ), where A047863(n) = Sum_{k=0..n} binomial(n, k) * (2^k)^(n-k).

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A001831 Number of labeled graded partially ordered sets with n elements of height at most 1.

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A001832 Number of labeled connected bipartite graphs on n nodes.

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A002031 Number of labeled connected digraphs on n nodes where every node has indegree 0 or outdegree 0 and no isolated nodes.

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A191371 Number of simple labeled graphs with (at most) 3-colored nodes such that no edge connects two nodes of the same color.

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A047864 Number of labeled bipartite graphs with n nodes.

A251684 G.f.: exp( Sum_{n>=1} A047863(n)x^n/n ), where A047863(n) = Sum_{k=0..n} binomial(n, k) (2^k)^(n-k).