The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A002027

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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.
(history; published version)

#28 by Alois P. Heinz at Mon Dec 03 18:27:35 EST 2018

STATUS

proposed

approved

#27 by Andrew Howroyd at Mon Dec 03 16:16:12 EST 2018

STATUS

editing

proposed

#26 by Andrew Howroyd at Mon Dec 03 16:15:24 EST 2018

FORMULA

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

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

#25 by Andrew Howroyd at Mon Dec 03 13:57:01 EST 2018

NAME

Colored 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.

LINKS

Andrew Howroyd, <a href="/A002027/b002027.txt">Table of n, a(n) for n = 0..50</a>

CROSSREFS

Column k=2 of A322279.

Essentially the same as A002031. Cf. A002032.

Cf. A002032.

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Name clarified by Andrew Howroyd, Dec 03 2018

STATUS

approved

editing

#24 by Joerg Arndt at Fri Jun 30 01:50:57 EDT 2017

STATUS

proposed

approved

#23 by Michel Marcus at Fri Jun 30 01:50:26 EDT 2017

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editing

proposed

#22 by Michel Marcus at Fri Jun 30 01:50:20 EDT 2017

REFERENCES

Read, R. C.; Wright, E. M.; Colored graphs: A correction and extension. Canad. J. Math. 22 1970 594-596.

LINKS

R. C. Read, RE. CM.; Wright, E. M., <a href="http://dx.doi.org/10.4153/CJM-1970-066-1">Colored graphs: A correction and extension</a>, Canad. J. Math. 22 1970 594-596.

STATUS

approved

editing

#21 by Joerg Arndt at Fri Sep 06 02:13:11 EDT 2013

STATUS

proposed

approved

#20 by Geoffrey Critzer at Thu Sep 05 16:56:01 EDT 2013

STATUS

editing

proposed

#19 by Geoffrey Critzer at Thu Sep 05 16:55:02 EDT 2013

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

FORMULA

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

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 *)

STATUS

approved

editing