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A014500
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Number of graphs with unlabeled (non-isolated) nodes and n labeled edges.
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7
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1, 1, 2, 9, 70, 794, 12055, 233238, 5556725, 158931613, 5350854707, 208746406117, 9315261027289, 470405726166241, 26636882237942128, 1678097862705130667, 116818375064650241036, 8932347052564257212796
(list;
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refs;
listen;
history;
text;
internal format)
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OFFSET
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0,3
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REFERENCES
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Cameron, Peter; Prellberg, Thomas; and Stark, Dudley; Asymptotic enumeration of 2-covers and line graphs. Discrete Math. 310 (2010), no. 2, 230-240 (see u_n).
G. Paquin, D\'enombrement de multigraphes enrichis, M\'emoire, Math. Dept., Univ. Qu\'ebec \`a Montr\'eal, 2004.
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LINKS
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Table of n, a(n) for n=0..17.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
G. Labelle, Counting enriched multigraphs..., Discrete Math., 217 (2000), 237-248.
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FORMULA
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E.g.f.: exp(-1+x/2)*Sum((1+x)^binomial(n, 2)/n!, n=0..infinity) [probably in the Labelle paper] - Vladeta Jovovic, Apr 27 2004
E.g.f.: exp(x/2)*Sum(A020556(n)*(ln(1+x)/2)^n/n!, n=0..infinity). - Vladeta Jovovic, May 02 2004
Binomial transform of A060053.
The e.g.f.'s of A020554 (S(x)) and A014500 (U(x)) are related by S(x) = U(e^x-1).
The e.g.f.'s of A014500 (U(x)) and A060053 (V(x)) are related by U(x) = e^x*V(x).
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MAPLE
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read("transforms") ;
A020556 := proc(n) local k; add((-1)^(n+k)*binomial(n, k)*combinat[bell](n+k), k=0..n) end proc:
A014500 := proc(n) local i, gexp, lexp;
gexp := [seq(1/2^i/i!, i=0..n+1)] ;
lexp := add( A020556(i)*((log(1+x))/2)^i/i!, i=0..n+1) ;
lexp := taylor(lexp, x=0, n+1) ;
lexp := gfun[seriestolist](lexp, 'ogf') ;
CONV(gexp, lexp) ; op(n+1, %)*n! ; end proc:
seq(A014500(n), n=0..20) ; # R. J. Mathar, Jul 03 2011
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CROSSREFS
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Cf. A020554, A020555, A014501, A060053.
Sequence in context: A201862 A167016 A108522 * A101482 A099717 A177450
Adjacent sequences: A014497 A014498 A014499 * A014501 A014502 A014503
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KEYWORD
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nonn
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AUTHOR
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Simon Plouffe, gilbert(AT)lacim.uqam.ca (Gilbert Labelle).
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STATUS
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approved
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