A002028 Number of connected graphs on n labeled nodes, each node being colored with one of 3 colors, such that no edge joins nodes of the same color.

1, 3, 6, 42, 618, 15990, 668526, 43558242, 4373213298, 677307561630, 162826875512646, 61183069270120842, 36134310487980825258, 33673533885068169649830, 49646105434209446798290206, 116002075479856331220877149042, 430053223599741677879550609246498, 2531493110297317758855120762121050990

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=3 of A322279.

Cf. A002031, A002032.

Programs

Mathematica

f[{k_, r_, m_}]:= Binomial[m+r+k, k] Binomial[m+r, r] 2^(k r +k m + r m);
  a = Sum[Total[Map[f, Compositions[n, 3]]] x^n/n!, {n, 0, 20}];
  Range[0, 20]! CoefficientList[Series[Log[a]+1, {x, 0, 20}], x] (* Geoffrey Critzer, Jun 02 2011 *)

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))^3))))} \\ Andrew Howroyd, Dec 03 2018

Formula

E.g.f.: log(A(x))+1 where A(x) is the e.g.f. for A191371. - Geoffrey Critzer, Jun 02 2011

a(n) = m_n(3) using the functions defined in A002032. - Sean A. Irvine, May 29 2013

Logarithmic transform of A191371. - Andrew Howroyd, Dec 03 2018