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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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.

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(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 *)

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

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Last modified January 20 04:43 EST 2019. Contains 319323 sequences. (Running on oeis4.)