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. (Formerly M3911 N1606)

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

Offset

0,2

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.

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

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 *)

Prog

(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

Crossrefs

Column k=5 of A322279.

Cf. A002032.

Sequence in context: A203213 A203206 A032072 * A213148 A202860 A344709

Adjacent sequences: A002027 A002028 A002029 * A002031 A002032 A002033

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Sean A. Irvine, May 27 2013

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

Status

approved