

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.
(Formerly M0365 N0138)


4



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

0,2


COMMENTS

a(n) is the number of connected labeled graphs with n 2colored 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 594596.


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


MATHEMATICA

nn=10; f[x_]:=Sum[Sum[Binomial[n, k]2^(k(nk)), {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 *)


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


CROSSREFS

Column k=2 of A322279.
Essentially the same as A002031.
Cf. A002032.
Sequence in context: A180069 A032185 A179236 * A290957 A032117 A137244
Adjacent sequences: A002024 A002025 A002026 * A002028 A002029 A002030


KEYWORD

nonn


AUTHOR

N. J. A. Sloane


EXTENSIONS

Corrected and extended by Sean A. Irvine, May 29 2013
Name clarified by Andrew Howroyd, Dec 03 2018


STATUS

approved



