A002030 - OEIS

%I M3911 N1606 #37 Sep 04 2019 10:23:51

%S 1,5,20,300,9980,616260,65814020,11878194300,3621432947180,

%T 1880516646144660,1678121372919602420,2590609089652498130700,

%U 6947580541943715645962780,32448510765823652400410879460,264301377639329321236008592510820

%N Number of connected graphs on n labeled nodes, each node being colored with one of 5 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="/A002030/b002030.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) where B(x) = Sum_{n>=0} b(n)x^n/n! and b(n) = Sum_{j=0..n} C(n, j)*2^(j*(n-j)+2)*A000686(j). - _Sean A. Irvine_, May 27 2013

%F a(n) = m_n(5) using the functions defined in A002032. - _Sean A. Irvine_, May 29 2013

%t m = 15;

%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)^5, 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))^5))))} \\ _Andrew Howroyd_, Dec 03 2018

%Y Column k=5 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