A000683 -id:A000683 - cp's OEIS frontend

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

A000684 Number of colored labeled n-node graphs with 2 interchangeable colors.

Original entry on oeis.org

1, 3, 13, 81, 721, 9153, 165313, 4244481, 154732801, 8005686273, 587435092993, 61116916981761, 9011561121239041, 1882834327457349633, 557257804202631217153, 233610656002563147038721, 138681207656726645785559041

Offset: 1

N. J. A. Sloane

a(n) = A058872(n) + 1. This sequence counts the empty graph on n nodes which is not allowed in A058872. - Geoffrey Critzer, Oct 07 2012

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).

Vincenzo Librandi, Table of n, a(n) for n = 1..100 (first 32 terms from R. W. Robinson)
S. R. Finch, Bipartite, k-colorable and k-colored graphs (2*A000684)
S. R. Finch, Bipartite, k-colorable and k-colored graphs, June 5, 2003. [Cached copy, with permission of the author]
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.]
R. C. Read, Letter to N. J. A. Sloane, Oct. 29, 1976

2 * A000683(n) + 1.

Cf. A058872, A111636, A134531.

With[{nn=20},Rest[CoefficientList[Series[Sum[x^n/(1-2^n x)^n,{n,nn}],{x,0,nn}], x]]] (* Harvey P. Dale, Nov 24 2011 *)

a(n)=polcoeff(sum(k=1,n,x^k/(1-2^k*x +x*O(x^n))^k),n) \\ Paul D. Hanna, Sep 14 2009

G.f.: A(x) = Sum_{n>=1} x^n/(1 - 2^n*x)^n. - Paul D. Hanna, Sep 14 2009
G.f.: 1/(W(0)-x) where W(k) = x*(x*2^k-1)^k - (x*2^(k+1)-1)^(k+1) + x*((2*x*2^k-1)^(2*k+2))/W(k+1); (continued fraction, Euler's 1st kind, 1-step). - Sergei N. Gladkovskii, Sep 17 2012
From Peter Bala, Apr 01 2013: (Start)
a(n) = Sum_{k = 0..n-1} binomial(n-1,k)*2^(k*(n-k)).
a(n) = Sum_{k = 0..n} 2^k*A111636(n,k).
Let E(x) = Sum_{n >= 0} x^n/(n!*2^C(n,2)). Then a generating function for this sequence (but with an offset of 0) is E(x)*E(2*x) = Sum_{n >= 0} a(n+1)*x^n/(n!*2^C(n,2)) = 1 + 3*x + 13*x^2/(2!*2) + 81*x^3/(3!*2^3) + 721*x^4/(4!*2^6) + .... Cf. A134531. (End)

a(15) onwards added by N. J. A. Sloane, Oct 19 2006 from the Robinson reference

A058843 Triangle T(n,k) = C_n(k) 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, 2, 1, 12, 8, 1, 80, 192, 64, 1, 720, 5120, 5120, 1024, 1, 9152, 192000, 450560, 245760, 32768, 1, 165312, 10938368, 56197120, 64225280, 22020096, 2097152, 1, 4244480, 976453632, 10877927424, 23781703680, 15971909632, 3758096384

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).
In this triangle, colorings of a labeled graph using k colors that differ only by a permutation of the k colors are treated as the same giving 1/k!*(E(x) - 1)^k as a generating function function for the k-th column. (End)

			Triangle begins:
  1;
  1,    2;
  1,   12,      8;
  1,   80,    192,     64;
  1,  720,   5120,   5120,   1024;
  1, 9152, 192000, 450560, 245760, 32768;
  ...

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
R. C. Read, The number of k-colored graphs on labelled nodes, Canad. J. Math., 12 (1960), 410-414.

Apart from scaling, same as A058875.
Columns give A058872 and A000683, A058873 and A006201, A058874 and A006202, also A006218. A213441, A213442, A224068.
Row sums give A240936.

for p from 1 to 20 do C[p,1] := 1; od: for k from 2 to 20 do for p from 1 to k-1 do C[p,k] := 0; od: od: for k from 2 to 10 do for p from k to 10 do C[p,k] := add( binomial(p,n)*2^(n*(p-n))*C[n,k-1]/k,n=1..p-1); od: od:

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], {n, 1, maxn}, {k, 1, n}]] (* Jean-François Alcover, Sep 21 2011 *)

T(n,k)={n!*2^binomial(n,2)*polcoef((sum(j=1, n, x^j/(j!*2^binomial(j,2))) + O(x*x^n))^k, n)/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) = sum {i = 1..n-1} binomial(n-1,i)*2^(i*(n-i))*T(i,k-1).
A generating function: exp(x*(E(z) - 1)) = 1 + x*z + (x + 2*x^2)*z^2/(2!*2) + (x + 12*x^2 + 8*x^3)*z^3/(3!*2^3) + .... Cf. A008277 with e.g.f. exp(x*(exp(z) - 1)).
A generating function for column k: 1/k!*(E(x) - 1)^k = sum {n>=k} T(n,k)x^n/(n!*2^C(n,2)).
The row polynomials R(n,x) satisfy the recurrence equation R(n,x) = x*(1 + sum {k = 0..n-1} binomial(n-1,k)*2^(k*(n-k))*R(k,x)) with R(1,x) = x. The row polynomials appear to have only real zeros.
Column 2 = 1/2!*A213441; Column 3 = 1/3!*A213442; Column 4 = 1/4!*A224068. (End)

A006202 Number of colorings of labeled graphs on n nodes using exactly 4 colors, divided by 4!*2^6.

Original entry on oeis.org

0, 0, 0, 1, 80, 7040, 878080, 169967616, 53247344640, 27580935700480, 23884321532149760, 34771166607668412416, 85316631064301031915520, 353171748158258855521812480, 2467057266045387831319241687040, 29078599995993904385498084987109376

Offset: 1

N. J. A. Sloane

Equals 1/1536*A224068. - Peter Bala, Apr 12 2013

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 18, col. 4 of Table 1.5.1 (divided by 64).
R. C. Read, personal communication.
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 = 1..80
R. C. Read, The number of k-colored graphs on labelled nodes, Canad. J. Math., 12 (1960), 410-414.
R. C. Read, Letter to N. J. A. Sloane, Oct. 29, 1976

A diagonal of A058875.

Cf. A000683, A224068.

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, 4]/64;
Array[a, maxn]

seq(n)={Vec(serconvol(sum(j=1, n, x^j*j!*2^binomial(j,2)) + O(x*x^n), (sum(j=1, n, x^j/(j!*2^binomial(j,2))) + O(x*x^n))^4)/1536, -n)} \\ Andrew Howroyd, Nov 30 2018

A046860 Triangle giving a(n,k) = number of k-colored labeled graphs with n nodes.

Original entry on oeis.org

1, 1, 4, 1, 24, 48, 1, 160, 1152, 1536, 1, 1440, 30720, 122880, 122880, 1, 18304, 1152000, 10813440, 29491200, 23592960, 1, 330624, 65630208, 1348730880, 7707033600, 15854469120, 10569646080, 1, 8488960, 5858721792, 261070258176, 2853804441600, 11499774935040, 18940805775360, 10823317585920

Offset: 1

N. J. A. Sloane

			Triangle begins:
  1;
  1,     4;
  1,    24,      48;
  1,   160,    1152,     1536;
  1,  1440,   30720,   122880,   122880;
  1, 18304, 1152000, 10813440, 29491200, 23592960;
  ...

Alois P. Heinz, Rows n = 1..50, flattened
R. C. Read, The number of k-colored graphs on labelled nodes, Canad. J. Math., 12 (1960), 410-414.

Column #1 gives A000683.
Main diagonal gives A011266.
Row sums give A334282.
Cf. A000683, A006201, A006202.

a:= proc(n, k) option remember; `if`([n, k]=[0$2], 1,
      add(binomial(n, r)*2^(r*(n-r))*a(r, k-1), r=0..n-1))
    end:
seq(seq(a(n,k), k=1..n), n=1..8);  # Alois P. Heinz, Apr 21 2020

a[n_ /; n >= 1, k_ /; k >= 1] := a[n, k] = Sum[ Binomial[n, r]*2^(r*(n - r))*a[r, k - 1], {r, 1, n - 1}]; a[, 0] = 1; Flatten[ Table[ a[n, k], {n, 1, 8}, {k, 0, n - 1}]] (* _Jean-François Alcover, Dec 12 2011, after formula *)

a(n, k) = Sum_{r=1..n-1} C(n, r) 2^(r*(n-r)) a(r, k-1).

1 + Sum_{n>=1} Sum_{k=1..n} a(n,k)*y^k*x^n/(n!*2^C(n,2)) = 1/(1-y(E(x)-1)) where E(x) = Sum_{n>=0} x^n/(n!*2^C(n,2)). - Geoffrey Critzer, May 06 2020

More terms from Vladeta Jovovic, Feb 04 2000

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

A006201 Number of colorings of labeled graphs on n nodes using exactly 3 colors, divided by 48.

Original entry on oeis.org

0, 0, 1, 24, 640, 24000, 1367296, 122056704, 17282252800, 3897054412800, 1400795928395776, 802530102499344384, 732523556206878392320, 1064849635418836398243840, 2464403435614136308036796416

Offset: 1

N. J. A. Sloane

Equals 1/48*A213442. - Peter Bala, Apr 12 2013

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 18, table 1.5.1, column 3 (divided by 8).
R. C. Read, personal communication.
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 = 1..75
R. C. Read, The number of k-colored graphs on labelled nodes, Canad. J. Math., 12 (1960), 410-414.
R. C. Read, Letter to N. J. A. Sloane, Oct. 29, 1976

Cf. A000683. A diagonal of A058843. A213442.

F2[n_] := Sum[Binomial[n, r]*2^(r*(n-r)), {r, 1, n-1}]; F3[n_] := Sum[Binomial[n, r]*2^(r*(n-r))*F2[r], {r, 1, n-1}]; a[n_] := F3[n]/48; Table[a[n], {n, 1, 15}] (* Jean-François Alcover, Mar 06 2014, after Maple code in A213442 *)

seq(n)={Vec(serconvol(sum(j=1, n, x^j*j!*2^binomial(j,2)) + O(x*x^n), (sum(j=1, n, x^j/(j!*2^binomial(j,2))) + O(x*x^n))^3)/48, -n)} \\ Andrew Howroyd, Nov 30 2018

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 1/48*(E(x) - 1)^3 = x^3/(3!*2^3) + 24*x^4/(4!*2^6) + 640*x^6/(5!*2^10) + ... (see Read). - Peter Bala, Apr 12 2013

More terms from Vladeta Jovovic, Feb 03 2000

A000685 Number of 3-colored labeled graphs on n nodes, divided by 3.

Original entry on oeis.org

1, 5, 41, 545, 11681, 402305, 22207361, 1961396225, 276825510401, 62368881977345, 22413909724518401, 12840603873823473665, 11720394922432296755201, 17037597932370037286600705

Offset: 1

N. J. A. Sloane

Sequence represents 1/3 of the number of 3-colored labeled graphs on n nodes. Indeed, on p. 413 of the Read paper, column 3 is 3, 15, 123, 1635, ...; or see A047863. - Emeric Deutsch, May 06 2004

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
S. R. Finch, Bipartite, k-colorable and k-colored graphs (3*A000685)
S. 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. C. Read, Letter to N. J. A. Sloane, Oct. 29, 1976
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

Cf. A000683, A047863, A191371, A223887.

c[0]:=1: for n from 1 to 30 do c[n]:=sum(binomial(n,i)*2^(i*(n-i)),i=0..n) od: a:=n->(1/3)*sum(binomial(n,j)*2^(j*(n-j))*c[j],j=0..n): seq(a(n),n=1..19);

a[n_] := 1/3*Sum[ 2^((i-j)*j + i*(n-i))*Binomial[n, i]*Binomial[i, j], {i, 0, n}, {j, 0, i}]; Table[ a[n], {n, 1, 14}] (* Jean-François Alcover, Dec 07 2011, after Emeric Deutsch *)

a(n) = (1/3)Sum_{j=0..n} binomial(n, j)*2^(j(n-j))*c(j) where c(n) = Sum_{i=0..n} binomial(n, i)*2^(i(n-i)) = A047863(n). - Emeric Deutsch, May 06 2004
From Peter Bala, Apr 12 2013: (Start)
a(n) = 1/3*A191371(n). Let E(x) = Sum_{n >= 0} x^n/(n!*2^C(n,2)). Then a generating function for this sequence is 1/3*E(x)^3 - 1/3 = Sum_{n >= 1} a(n)*x^n/(n!*2^C(n,2)) = x + 5*x^2/(2!*2) + 41*x^3/(3!*2^3) + .... In general, E(x)^k, k = 1, 2, ..., is a generating function for labeled k-colored graphs (see Read). For examples see A047863 (k = 2), A191371 (k = 3) and A223887 (k = 4). (End)

More terms from Pab Ter (pabrlos(AT)yahoo.com) and Emeric Deutsch, May 05 2004

A000686 Number of 4-colored labeled graphs on n nodes, divided by 4.

Original entry on oeis.org

1, 7, 85, 1777, 63601, 3882817, 403308865, 71139019777, 21276992674561, 10778161937857537, 9238819435213784065, 13390649605615389843457, 32796747486424209782108161, 135669064080920007649863745537, 947468281528010179181982467702785, 11166618111585805201637975219611631617

Offset: 1

N. J. A. Sloane

Sequence represents 1/4 of the number of 4-colored labeled graphs on n nodes. Indeed, on p. 413 of the Read paper, column 4 is 4, 28, 340, 7108, ... - Emeric Deutsch, May 06 2004

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).

S. R. Finch, Bipartite, k-colorable and k-colored graphs. (4*{a(n)})
S. 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. C. Read, Letter to N. J. A. Sloane, Oct. 29, 1976.

Cf. A000683, A000685, A223887.

b[n_] := Sum[ 2^((i-j)*j + i*(n-i))*Binomial[n, i]*Binomial[i, j], {i, 0, n}, {j, 0, i}]; a[n_] := 1/4*Sum[ Binomial[n, k]*2^(k*(n-k))*b[k], {k, 0, n}]; Table[a[n], {n, 1, 14}] (* Jean-François Alcover, Dec 07 2011, after Emeric Deutsch *)

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

a(n) = (1/4)*Sum_{k=0..n} binomial(n, k)*2^(k(n-k))*b(k), where b(0)=1 and b(k) = 3*A000685(k) for k > 0. - Emeric Deutsch, May 06 2004
From Peter Bala, Apr 12 2013: (Start)
a(n) = (1/4)*A223887(n).
a(n) = (1/4)*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 (1/4)*(E(x)^4 - 1) = Sum_{n >= 0} a(n)*x^n/(n!*2^C(n,2)) = x + 7*x^2/(2!*2) + 85*x^3/(3!*2^3) + .... (End)

More terms from Pab Ter (pabrlos(AT)yahoo.com) and Emeric Deutsch, May 05 2004

A052263 Number of 5-colored labeled graphs on n nodes (divided by 1024).

Original entry on oeis.org

0, 0, 0, 0, 1, 240, 62720, 23224320, 13440516096, 12754259804160, 20377681883299840, 55671587500342640640, 262483693581093123915776, 2147534811240814766425374720, 30585643102219392981931508367360

Offset: 1

Vladeta Jovovic, Feb 03 2000

R. C. Read, The number of k-colored graphs on labelled nodes, Canad. J. Math., 12 (1960), 410-414.

Cf. A000683, A006201, A006202.

a(n, k) = (1/k)*Sum_{r=1..n-1} C(n, r) 2^(r*(n-r)) a(r, k-1).

cp's OEIS Frontend

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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A000684 Number of colored labeled n-node graphs with 2 interchangeable colors.

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A058843 Triangle T(n,k) = C_n(k) where C_n(k) = number of k-colored labeled graphs with n nodes (n >= 1, 1<=k<=n).

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A006202 Number of colorings of labeled graphs on n nodes using exactly 4 colors, divided by 4!*2^6.

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A046860 Triangle giving a(n,k) = number of k-colored labeled graphs with n nodes.

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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).

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