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Number of connected graphs on n labeled nodes, each node being colored with one of 4 colors, such that no edge joins nodes of the same color.
(Formerly M3459 N1406)
3

%I M3459 N1406 #36 Sep 05 2019 00:04:46

%S 1,4,12,132,3156,136980,10015092,1199364852,234207001236,

%T 75018740661780,39745330657406772,35073541377640231092,

%U 51798833078501480220756,128412490016744675540378580,535348496386845235339961362932,3757366291145650829115977555259252

%N Number of connected graphs on n labeled nodes, each node being colored with one of 4 colors, such that no edge joins nodes of the same color.

%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).

%D R. C. Read, personal communication.

%H Andrew Howroyd, <a href="/A002029/b002029.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(b(x)+1)+1 where b(x) = 4 * e.g.f. of A000686. - _Sean A. Irvine_, May 27 2013

%F a(n) = m_n(4) using the functions defined in A002032. - _Sean A. Irvine_, May 29 2013

%F Logarithmic transform of A223887. - _Andrew Howroyd_, Dec 03 2018

%t m = 16;

%t serconv = (CoefficientList[Sum[x^j*2^Binomial[j, 2], {j, 0, m}] + O[x]^m, x]*CoefficientList[(Sum[x^j/(j!*2^Binomial[j, 2]), {j, 0, m}] + O[x]^m)^4, x]) . x^Range[0, m-1];

%t CoefficientList[1 + Log[serconv] + O[x]^m, x]*Range[0, m-1]! (* _Jean-François Alcover_, Sep 04 2019, after _Andrew Howroyd_ *)

%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))^4))))} \\ _Andrew Howroyd_, Dec 03 2018

%Y Column k=4 of A322279.

%Y Cf. A002032.

%K nonn

%O 0,2

%A _N. J. A. Sloane_

%E More terms from _Sean A. Irvine_, May 27 2013

%E Name clarified and offset corrected by _Andrew Howroyd_, Dec 03 2018