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

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

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

References

• 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).

Links

• 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.

Crossrefs

a(n)=(A047863(n)-2)/4.

A diagonal of A058843.

One quarter of A213441.

Programs

Mathematica

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 *)

Formula

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

Extensions

More terms from Vladeta Jovovic, Feb 02 2000