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A002027
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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)
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5
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1, 2, 2, 6, 38, 390, 6062, 134526, 4172198, 178449270, 10508108222, 853219059726, 95965963939958, 15015789392011590, 3282145108526132942, 1005193051984479922206, 432437051675617901246918, 261774334771663762228012950, 223306437526333657726283273822
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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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.
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FORMULA
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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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
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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Corrected and extended by Sean A. Irvine, May 29 2013
Name clarified by Andrew Howroyd, Dec 03 2018
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STATUS
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approved
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