A047863 -id:A047863 - cp's OEIS frontend

A135079 E.g.f. A(x) = Sum_{n>=0} exp(3^nx)x^n/n!.

1, 2, 8, 56, 704, 15392, 593408, 39691136, 4650143744, 944100803072, 334651494268928, 205435333440321536, 219775256161359233024, 407034554694060677537792, 1312205966809501720566038528

Offset: 0

Paul D. Hanna, Nov 24 2007

a(n) is the number of labeled graphs with (at most) 2 colors of vertices where vertices of the same color are never adjacent and the graphs may have up to 2 types of edges. - Geoffrey Critzer, Apr 20 2020

G. C. Greubel, Table of n, a(n) for n = 0..75

Cf. A047863 (variant). A188457.

Table[Sum[Binomial[n,k]*3^(k*(n-k)),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jun 24 2013 *)

{a(n)=sum(k=0,n,binomial(n,k)*3^(k*(n-k)))}

/* E.g.f.: */ {a(n)=n!*polcoeff(sum(k=0,n,exp(3^k*x +x*O(x^n))*x^k/k!),n)}

{a(n)=polcoeff(sum(k=0,n,x^k/(1 - 3^k*x +x*O(x^n))^(k+1)),n)} \\ Paul D. Hanna, Aug 08 2009

a(n) = Sum_{k=0..n} C(n, k)*3^(k*(n-k)).
O.g.f.: A(x) = Sum_{n>=0} x^n/(1 - 3^n*x)^(n+1). - Paul D. Hanna, Aug 08 2009
Let E(x) = sum {n >= 0} x^n/(n!*3^C(n,2)). Then a generating function for this sequence is E(x)^2 = sum {n >= 0} a(n)*x^n/(n!*3^C(n,2)) = 1 + 2*x + 8*x^2/(2!*3) + 56*x^3/(3!*3^3) + 704*x^4/(4!*3^6) + .... Cf. A188457. - Peter Bala, Apr 01 2013
a(n) ~ c * 3^(n^2/4)*2^(n+1/2)/sqrt(Pi*n), where c = Sum_{k = -infinity..infinity} 3^(-k^2) = 1.6914596816817... if n is even and c = Sum_{k = -infinity..infinity} 3^(-(k+1/2)^2) = 1.69061120307521... if n is odd. - Vaclav Kotesovec, Jun 24 2013

A052332 Number of labeled digraphs where every node has indegree 0 or outdegree 0 and no isolated nodes.

Original entry on oeis.org

1, 0, 2, 6, 50, 510, 7682, 161406, 4747010, 194342910, 11084390402, 881008805886, 97779099906050, 15178191426486270, 3302331237256396802, 1008694542117649154046, 433286992912494943469570

Offset: 0

Christian G. Bower, Dec 15 1999

nonn

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

Cf. A001831 (binomial transform), A002031, A047863.

a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*A001831(k)

a(n) = Sum_{k=0..n} binomial(n, k)*(2^k-2)^(n-k). - Vladeta Jovovic, Apr 04 2003

Last 4 terms corrected by Vladeta Jovovic, Apr 04 2003

A223887 Number of 4-colored labeled graphs on n vertices.

Original entry on oeis.org

1, 4, 28, 340, 7108, 254404, 15531268, 1613235460, 284556079108, 85107970698244, 43112647751430148, 36955277740855136260, 53562598422461559373828, 131186989945696839128432644, 542676256323680030599454982148

Offset: 0

Peter Bala, Apr 10 2013

A simple graph G is a k-colorable graph if it is possible to assign one of k' <= k colors to each vertex of G so that no two adjacent vertices receive the same color. Such an assignment of colors is called a coloring function for the graph.
A k-colored graph is a k-colorable graph together with its coloring function. This sequence gives the number of labeled 4-colored graphs on n vertices. An example is given below.
See A047863 for labeled 2-colored graphs on n vertices and A191371 for labeled 3-colored graphs on n vertices. See A076316 for labeled 4-colorable graphs on n vertices and A224068 for the count of labeled graphs colored using exactly 4 colors.

			a(2) = 28: There are two labeled 4-colorable graphs on 2 nodes, namely
A)  1    2  B)  1    2
    o    o      o----o
Using 4 colors there are 16 ways to color the graph of type A and 4*3 = 12 ways to color the graph of type B so that adjacent vertices do not share the same color. Thus there are in total 28 labeled 4-colored graphs on 2 vertices.

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

Andrew Howroyd, Table of n, a(n) for n = 0..50
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.
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=4 of A322280.

Equals 4 * A000686, A047863 (labeled 2-colored graphs), A076316, A191371 (labeled 3-colored graphs), A224068.

N=66;  x='x+O('x^N);
E=sum(n=0, N, x^n/(n!*2^binomial(n,2)) );
tgf=E^4;  v=concat(Vec(tgf));
v=vector(#v, n, v[n] * (n-1)! * 2^((n-1)*(n-2)/2) )
/* Joerg Arndt, Apr 10 2013 */

a(n) = sum {k = 0..n} binomial(n,k)*2^(k*(n-k))*b(k)*b(n-k), where b(n) := sum {k = 0..n} binomial(n,k)*2^(k*(n-k)).

Let E(x) = sum {n >= 0} x^n/(n!*2^C(n,2)). Then a generating function for this sequence is E(x)^4 = sum {n >= 0} a(n)*x^n/(n!*2^C(n,2)) = 1 + 4*x + 28*x^2/(2!*2) + 340*x^3/(3!*2^3) + .... In general, for k = 1, 2, ..., E(x)^k is a generating function for labeled k-colored graphs (see Stanley).

A111636 Triangle read by rows: T(n,k) (0<=k<=n) is the number of labeled graphs having k blue nodes and n-k green ones and only nodes of different colors can be joined by an edge.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 12, 12, 1, 1, 32, 96, 32, 1, 1, 80, 640, 640, 80, 1, 1, 192, 3840, 10240, 3840, 192, 1, 1, 448, 21504, 143360, 143360, 21504, 448, 1, 1, 1024, 114688, 1835008, 4587520, 1835008, 114688, 1024, 1, 1, 2304, 589824, 22020096, 132120576, 132120576, 22020096, 589824, 2304, 1

Offset: 0

Emeric Deutsch, Aug 09 2005

Row sums yield A047863. T(2*n,n) = A111637(n). T(n,1) = A001787(n).

			T(2,1)=4 because we have B G, B--G, G B and G--B, where B (G) stands for a blue (green) node and -- denotes an edge.
Triangle starts:
  1;
  1,  1;
  1,  4,  1;
  1, 12, 12,  1;
  1, 32, 96, 32, 1;
  ...

H. S. Wilf, Generatingfunctionology, 2nd edn., Academic Press, NY, 1994, p. 88, Eq. 3.11.2.

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]
W. Wang and T. Wang, Generalized Riordan array, Discrete Mathematics, Vol. 308, No. 24, 6466-6500.

Cf. A047863, A111637, A001787.
Cf. A134530 (matrix log), A134531.
Cf. A000684, A011266, A038845, A140802, A224069 (matrix inverse).

T:=(n,k)->binomial(n,k)*2^(k*(n-k)): for n from 0 to 9 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

nn=6;f[x_,y_]:=Sum[Exp[x 2^n](y x)^n/n!,{n,0,nn}];Range[0,nn]!CoefficientList[Series[f[x,y],{x,0,nn}],{x,y}]//Grid (* Geoffrey Critzer, Sep 07 2013 *)

T(n, k)=2^[k(n-k)]*C(n, k).
Matrix log yields triangle A134530, where A134530(n,k) = A134531(n-k)*(2^k)^(n-k)*C(n,k). - Paul D. Hanna, Nov 11 2007
From Peter Bala, Aug 13 2012: (Start)
Let f(n) = n!*2^binomial(n,2) = A011266(n). Then T(n,k) = f(n)/(f(k)*f(n-k)).
E.g.f.: Sum_{n >= 0} exp(2^n*t*x)*x^n/n! = 1 + (1+t)*x + (1+4*t+t^2)*x^2/2! + ....
O.g.f.: Sum_{n >= 0} x^n/(1-2^n*t*x)^(n+1) = 1 + (1+t)*x + (1+4*t+t^2)*x^2 + .... O.g.f. for column k: 1/(1-2^k*x)^(k+1).
Recurrence equation: T(n,k) = 2^k*T(n-1,k) + 2^(n-k)*T(n-1,k-1).
Column k = 2: A038845. Column k = 3: A140802. Sum_{k = 0..n} k*T(n,k) = n*A000684(n). (End)
From Peter Bala, Apr 09 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 for this sequence is E(z)*E(x*z) = 1 + (1 + x)*z + (1 + 4*x + x^2)*z^2/(2!*2) + (1 + 12*x + 12*x^2 + x^3)*z^3/(3!*2^3) + .... Cf. Pascal's triangle A007318 with an e.g.f. of exp(z)*exp(x*z).
This is a generalized Riordan array (E(x), x) with respect to the sequence n!*2^C(n,2), as defined by Wang and Wang.
The n-th power of this triangle has a generating function E(z)^n*E(x*z). See A224069 for the inverse array (n = -1).
The n-th row is a log-concave sequence and hence unimodal.
The row polynomials satisfy the recurrence equation R(n+1,x) = 2^n*x*R(n,x/2) + R(n,2*x) with R(0,x) = 1, as well as R'(n,2*x) = n*2^(n-1)*R(n-1,x) (the ' denotes differentiation w.r.t. x). The row polynomials appear to have only real zeros.
Sum_{k = 0..n} (-1)^k*T(2*n+1,k) = 0;
Sum_{k = 0..n} (-1)^k*2^k*T(2*n,k) = 0;
Sum_{k = 0..n} 2^k*T(n,k) = A000684(n). (End)
T(n,k+1) = Product_{i=0..k} (T(n-i,1)/T(i+1,1)) for 0 <= k < n. - Werner Schulte, Nov 13 2018

A117402 Row sums of triangle A117401: a(n) = Sum_{k=0..n} 2^((n-k)*k) for n>=0.

Original entry on oeis.org

1, 2, 4, 10, 34, 162, 1090, 10370, 139522, 2654722, 71435266, 2718435330, 146299424770, 11134711111682, 1198484887715842, 182431106853797890, 39271952800672710658, 11955805018770498519042, 5147453397489773531365378

Offset: 0

Paul D. Hanna, Mar 12 2006

nonn

a(n) is the number of 2-colored labeled graphs (as in A047863) such that the black nodes are labeled with {1,2,...,k} where k, 0<=k<=n, is the number of black nodes and the white nodes are labeled with {k+1,k+2,...,n}. These graphs form the desired binomial poset (for the case q=2) in the "task left to the reader" in the Stanley reference below. - Geoffrey Critzer, May 31 2020

			A(x) = 1/(1-x) + x/(1-2x) + x^2/(1-4x) + x^3/(1-8x) + ...

R. P. Stanley, Enumerative Combinatorics, Volume I, Second Edition, Cambridge, 2012, Example 3.18.3 e, page 323.

Robert Israel, Table of n, a(n) for n = 0..116

Cf. A117401 (triangle), A117403 (antidiagonal sums).

[(&+[2^(k*(n-k)): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Jun 28 2021

N:= 25:
G:= series(add(x^n/(1-2^n*x),n=0..N),x,N+1):
seq(coeff(G,x,n),n=0..N)); # Robert Israel, Dec 11 2018

a[n_]:= Sum[2^((n-k)*k), {k,0,n}]; Array[a, 20, 0] (* Amiram Eldar, Dec 12 2018 *)

PARI
```
a(n)=sum(k=0,n,2^((n-k)*k))
```

[sum(2^(k*(n-k)) for k in (0..n)) for n in (0..20)] # G. C. Greubel, Jun 28 2021

G.f.: A(x) = Sum_{n>=0} x^n/(1-2^n*x).
Let E(x) = Sum_{n>=0} x^n/2^C(n,2). Then E(x)^2 = Sum_{n>=0} a(n)*x^n/2^C(n,2). - Geoffrey Critzer, May 31 2020
a(n) ~ c * 2^(n^2/4), where c = EllipticTheta[3, 0, 1/2] = JacobiTheta3(0,1/2) = 2.128936827211877158669458548544951324612516539940878092889... if n is even and c = EllipticTheta[2, 0, 1/2] = JacobiTheta2(0,1/2) = 2.128931250513027558591613402575350180853805396958448940969... if n is odd. - Vaclav Kotesovec, Jun 28 2021

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

A361950 Array read by antidiagonals: T(n,k) = n! * Sum_{s} 2^(Sum_{i=1..k-1} s(i)*s(i+1))/(Product_{i=1..k} s(i)!) where the sum is over all nonnegative compositions s of n into k parts.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 6, 1, 0, 1, 4, 13, 26, 1, 0, 1, 5, 22, 81, 162, 1, 0, 1, 6, 33, 166, 721, 1442, 1, 0, 1, 7, 46, 287, 1726, 9153, 18306, 1, 0, 1, 8, 61, 450, 3309, 24814, 165313, 330626, 1, 0, 1, 9, 78, 661, 5650, 50975, 494902, 4244481, 8488962, 1, 0

Offset: 0

Andrew Howroyd, Mar 31 2023

T(n,k) corresponds to c(k,n) in the Klarner reference. This is an intermediate step in the computation of the number of labeled weakly graded (ranked) posets. The number of elements in the poset is n and the rank k.

			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      13       22       33       46 ...
3  | 0 1     26      81      166      287      450 ...
4  | 0 1    162     721     1726     3309     5650 ...
5  | 0 1   1442    9153    24814    50975    91866 ...
6  | 0 1  18306  165313   494902  1058493  1957066 ...
7  | 0 1 330626 4244481 13729846 29885567 55363650 ...
  ...
T(3,2) = 26: the nonnegative integer compositions of 3 with 2 parts are (0,3), (1,2), (2,1), (3,0). These contribute, respectively 2^0*3!/(0!*3!) = 1, 2^2*3!/(1!*2!) = 12, 2^2*3!/(2!*1!) = 12, 2^0*3!/(0!*3!) = 1, so T(3,2) = 1 + 12 + 12 + 1 = 26.

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals).
D. A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19.

Columns k=0..7 are A000007, A000012, A047863, A000684, A001827, A001828, A001829, A001830.
Rows 0..2 are A000012, A000027, A028872(n+1).
The unlabeled version is A361952.
Cf. A361951.

S(M)={matrix(#M, #M, i, j, sum(k=0, i-j, 2^((j-1)*k)*M[i-j+1,k+1])/(j-1)! )}
C(n, m=n)={my(M=matrix(n+1, n+1), c=vector(m+1), A=O(x*x^n)); M[1, 1]=1; c[1]=1+A; for(h=1, m, M=S(M); c[h+1]=sum(i=0, n, vecsum(M[i+1, ])*x^i, A)); c}
R(n)={Mat([Col(serlaplace(p)) | p<-C(n)])}
{ my(A=R(6)); for(i=1, #A, print(A[i,])) }

A001827 Related to graded partially ordered sets.

Original entry on oeis.org

1, 4, 22, 166, 1726, 24814, 494902, 13729846, 531077086, 28697950174, 2170176736102, 230007989092006, 34211282155446286, 7149766552058591374, 2101690590380890192342, 869808621195903097079446, 507261036269544624540347326

Offset: 0

N. J. A. Sloane

nonn

Corresponds to the numbers c(4,n) in the Klarner paper. - Sean A. Irvine, Sep 24 2015

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).

John Cerkan, Table of n, a(n) for n = 0..112
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.
Index entries for sequences related to posets

Column k=4 of A361950.

Cf. A000684, A001828, A001829, A001830, A047863.

a(n) = Sum_{p+q+r+s=n} (n!/p!q!r!s!) 2^(pq+qr+rs) where (p,q,r,s) is any nonnegative composition of n. - Sean A. Irvine, Sep 24 2015

More terms from Sean A. Irvine, Sep 24 2015

A001828 Related to graded partially ordered sets.

Original entry on oeis.org

1, 5, 33, 287, 3309, 50975, 1058493, 29885567, 1156711869, 61815727295, 4589058616413, 475576073939807, 69061902766811229, 14093318360697120095, 4049931601653596366013, 1641314561238334948886207, 939097032426474389539281789

Offset: 0

N. J. A. Sloane

nonn

Corresponds to the numbers c(5,n) in the Klarner paper. - Sean A. Irvine, Sep 24 2015

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).

John Cerkan, Table of n, a(n) for n = 0..112
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.
Index entries for sequences related to posets

Column k=5 of A361950.

Cf. A000684, A001827, A001829, A001830, A047863.

a(n) = Sum_{p+q+r+s+t=n} (n!/p!q!r!s!t!) 2^(pq+qr+rs+st) where (p,q,r,s,t) is any nonnegative composition of n. - Sean A. Irvine, Sep 24 2015

More terms from Sean A. Irvine, Sep 24 2015

A001829 Related to graded partially ordered sets.

Original entry on oeis.org

1, 6, 46, 450, 5650, 91866, 1957066, 55363650, 2109599650, 109773407466, 7894945079386, 792252362302770, 111671194813402930, 22202849561274787866, 6241728810901739517226, 2484011055161613143144610, 1400187830319472451472442690

Offset: 0

N. J. A. Sloane

nonn

Corresponds to the numbers c(6,n) in the Klarner paper. - Sean A. Irvine, Sep 24 2015

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).

John Cerkan, Table of n, a(n) for n = 0..112
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.
Index entries for sequences related to posets

Column k=6 of A361950.

Cf. A000684, A001827, A001828, A001830, A047863.

a(n) = Sum_{p+q+r+s+t+u=n} (n!/p!q!r!s!t!u!) 2^(pq+qr+rs+st+tu) where (p,q,r,s,t,u) is any nonnegative composition of n. - Sean A. Irvine, Sep 24 2015

cp's OEIS Frontend

A135079 E.g.f. A(x) = Sum_{n>=0} exp(3^n*x)*x^n/n!.

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A052332 Number of labeled digraphs where every node has indegree 0 or outdegree 0 and no isolated nodes.

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A223887 Number of 4-colored labeled graphs on n vertices.

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A111636 Triangle read by rows: T(n,k) (0<=k<=n) is the number of labeled graphs having k blue nodes and n-k green ones and only nodes of different colors can be joined by an edge.

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Maple

Mathematica

Formula

A117402 Row sums of triangle A117401: a(n) = Sum_{k=0..n} 2^((n-k)*k) for n>=0.

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Magma

Maple

Mathematica

PARI

Sage

Formula

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

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Maple

Mathematica

PARI

Formula

A361950 Array read by antidiagonals: T(n,k) = n! * Sum_{s} 2^(Sum_{i=1..k-1} s(i)*s(i+1))/(Product_{i=1..k} s(i)!) where the sum is over all nonnegative compositions s of n into k parts.

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A135079 E.g.f. A(x) = Sum_{n>=0} exp(3^nx)x^n/n!.