A002027 Number of connected graphs on n labeled nodes, each node being colored with one of 2 colors, such that no edge joins nodes of the same color.

1, 2, 2, 6, 38, 390, 6062, 134526, 4172198, 178449270, 10508108222, 853219059726, 95965963939958, 15015789392011590, 3282145108526132942, 1005193051984479922206, 432437051675617901246918, 261774334771663762228012950, 223306437526333657726283273822

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

a(n) is the number of connected labeled graphs with n 2-colored nodes where black nodes are only connected to white nodes and vice versa. - Geoffrey Critzer, Sep 05 2013

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=2 of A322279.

Essentially the same as A002031.

Cf. A002032.

Programs

Mathematica

nn=10;f[x_]:=Sum[Sum[Binomial[n,k]2^(k(n-k)),{k,0,n}]x^n/n!,{n,0,nn}];Range[0,nn]!CoefficientList[Series[Log[f[x]]+1,{x,0,nn}],x] (* Geoffrey Critzer, Sep 05 2013 *)

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))^2))))} \\ Andrew Howroyd, Dec 03 2018

Formula

a(n) = m_n(2) using the functions defined in A002032. - Sean A. Irvine, May 29 2013

E.g.f.: log(A(x))+1 where A(x) is the e.g.f. for A047863. - Geoffrey Critzer, Sep 05 2013

Logarithmic transform of A047863. - Andrew Howroyd, Dec 03 2018

Extensions

Corrected and extended by Sean A. Irvine, May 29 2013

Name clarified by Andrew Howroyd, Dec 03 2018