%I M4238 N1770 #37 Nov 25 2015 01:40:11

%S 0,1,6,40,360,4576,82656,2122240,77366400,4002843136,293717546496,

%T 30558458490880,4505780560619520,941417163728674816,

%U 278628902101315608576,116805328001281573519360

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

%C 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

%C A047863 counts colorings of labeled graphs on n vertices that use two or fewer colors. - _Peter Bala_, Apr 11 2013

%D F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 18, table 1.5.1, column 2 (divided by 2).

%D R. C. Read, personal communication.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A000683/b000683.txt">Table of n, a(n) for n=1..50</a>

%H R. C. Read, <a href="http://cms.math.ca/10.4153/CJM-1960-035-0">The number of k-colored graphs on labelled nodes</a>, Canad. J. Math., 12 (1960), 410—414.

%F Reference gives generating function.

%F 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

%t 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 *)

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

%Y A diagonal of A058843.

%Y One quarter of A213441.

%K nonn,nice,easy

%O 1,3

%A _N. J. A. Sloane_

%E More terms from _Vladeta Jovovic_, Feb 02 2000