A002030 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.

1, 5, 20, 300, 9980, 616260, 65814020, 11878194300, 3621432947180, 1880516646144660, 1678121372919602420, 2590609089652498130700, 6947580541943715645962780, 32448510765823652400410879460, 264301377639329321236008592510820

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

References

• R. C. Read, personal communication.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..50
• R. C. Read, E. M. Wright, Colored graphs: A correction and extension, Canad. J. Math. 22 1970 594-596.

Crossrefs

Column k=5 of A322279.

Cf. A002032.

Programs

Mathematica

m = 15;
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];
CoefficientList[1 + Log[serconv] + O[x]^m, x]*Range[0, m-1]! (* Jean-François Alcover, Sep 04 2019, after Andrew Howroyd *)

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

Formula

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

a(n) = m_n(5) using the functions defined in A002032. - Sean A. Irvine, May 29 2013

Extensions

More terms from Sean A. Irvine, May 27 2013

Name clarified and offset corrected by Andrew Howroyd, Dec 03 2018