A006202 -id:A006202 - cp's OEIS frontend

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).

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)

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

A058874 Number of 4-colored labeled graphs with n nodes.

Original entry on oeis.org

0, 0, 0, 64, 5120, 450560, 56197120, 10877927424, 3407830056960, 1765179884830720, 1528596578057584640, 2225354662890778394624, 5460264388115266042593280, 22602991882128566753395998720, 157891665026904821204431467970560

Offset: 1

N. J. A. Sloane, Jan 07 2001

nonn

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..80

A diagonal of A058843.

Equals 64 * A006202.

m = 16;
CoefficientList[Sum[x^j*j!*2^Binomial[j, 2], {j, 1, m}] + O[x]^n, x]* CoefficientList[(Sum[x^j/(j!*2^Binomial[j, 2]), {j, 1, n}] + O[x]^m)^4/24 + O[x]^m, x] // Rest (* Jean-François Alcover, Sep 06 2019, from PARI *)

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)/24, -n)} \\ Andrew Howroyd, Nov 30 2018

cp's OEIS Frontend

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).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

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).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A058874 Number of 4-colored labeled graphs with n nodes.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Links

Crossrefs

Formula