This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A000683 M4238 N1770 #44 May 11 2025 01:05:41
%S A000683 0,1,6,40,360,4576,82656,2122240,77366400,4002843136,293717546496,
%T A000683 30558458490880,4505780560619520,941417163728674816,
%U A000683 278628902101315608576,116805328001281573519360,69340603828363322892779520,58287619305053298399714082816,69366390252412220606233109200896
%N A000683 Number of colorings of labeled graphs on n nodes using exactly 2 colors, divided by 4.
%C A000683 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 A000683 A047863 counts colorings of labeled graphs on n vertices that use two or fewer colors. - _Peter Bala_, Apr 11 2013
%D A000683 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 18, table 1.5.1, column 2 (divided by 2).
%D A000683 R. C. Read, personal communication.
%D A000683 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000683 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000683 T. D. Noe, <a href="/A000683/b000683.txt">Table of n, a(n) for n=1..50</a>
%H A000683 R. C. Read, <a href="https://doi.org/10.4153/CJM-1960-035-0">The number of k-colored graphs on labelled nodes</a>, Canad. J. Math., 12 (1960), 410-414.
%F A000683 Reference gives generating function.
%F A000683 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 A000683 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 A000683 a(n)=(A047863(n)-2)/4.
%Y A000683 A diagonal of A058843.
%Y A000683 One quarter of A213441.
%K A000683 nonn,nice,easy
%O A000683 1,3
%A A000683 _N. J. A. Sloane_
%E A000683 More terms from _Vladeta Jovovic_, Feb 02 2000