A002029 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.

1, 4, 12, 132, 3156, 136980, 10015092, 1199364852, 234207001236, 75018740661780, 39745330657406772, 35073541377640231092, 51798833078501480220756, 128412490016744675540378580, 535348496386845235339961362932, 3757366291145650829115977555259252

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

References

• 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).
• R. C. Read, personal communication.

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=4 of A322279.

Cf. A002032.

Programs

Mathematica

m = 16;
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];
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))^4))))} \\ Andrew Howroyd, Dec 03 2018

Formula

E.g.f.: log(b(x)+1)+1 where b(x) = 4 * e.g.f. of A000686. - Sean A. Irvine, May 27 2013

a(n) = m_n(4) using the functions defined in A002032. - Sean A. Irvine, May 29 2013

Logarithmic transform of A223887. - Andrew Howroyd, Dec 03 2018

Extensions

More terms from Sean A. Irvine, May 27 2013

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