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Number of connected graphs on n labeled nodes, each node being colored with one of 3 colors, such that no edge joins nodes of the same color.
(Formerly M2603 N1030)
3

%I M2603 N1030 #35 Dec 03 2018 17:46:21

%S 1,3,6,42,618,15990,668526,43558242,4373213298,677307561630,

%T 162826875512646,61183069270120842,36134310487980825258,

%U 33673533885068169649830,49646105434209446798290206,116002075479856331220877149042,430053223599741677879550609246498,2531493110297317758855120762121050990

%N Number of connected graphs on n labeled nodes, each node being colored with one of 3 colors, such that no edge joins nodes of the same color.

%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 Andrew Howroyd, <a href="/A002028/b002028.txt">Table of n, a(n) for n = 0..50</a>

%H R. C. Read, E. M. Wright, <a href="http://dx.doi.org/10.4153/CJM-1970-066-1">Colored graphs: A correction and extension</a>, Canad. J. Math. 22 1970 594-596.

%F E.g.f.: log(A(x))+1 where A(x) is the e.g.f. for A191371. - _Geoffrey Critzer_, Jun 02 2011

%F a(n) = m_n(3) using the functions defined in A002032. - _Sean A. Irvine_, May 29 2013

%F Logarithmic transform of A191371. - _Andrew Howroyd_, Dec 03 2018

%t f[{k_, r_, m_}]:= Binomial[m+r+k, k] Binomial[m+r, r] 2^(k r +k m + r m);

%t a = Sum[Total[Map[f, Compositions[n, 3]]] x^n/n!, {n, 0, 20}];

%t Range[0, 20]! CoefficientList[Series[Log[a]+1, {x, 0, 20}], x] (* _Geoffrey Critzer_, Jun 02 2011 *)

%o (PARI) seq(n)={Vec(serlaplace(1 + log(serconvol(sum(j=0, n, x^j*2^binomial(j, 2)) + O(x*x^n), (sum(j=0, n, x^j/(j!*2^binomial(j, 2))) + O(x*x^n))^3))))} \\ _Andrew Howroyd_, Dec 03 2018

%Y Column k=3 of A322279.

%Y Cf. A002031, A002032.

%K nonn

%O 0,2

%A _N. J. A. Sloane_