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A001187
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Number of connected labeled graphs with n nodes.
(Formerly M3671 N1496)
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24
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1, 1, 1, 4, 38, 728, 26704, 1866256, 251548592, 66296291072, 34496488594816, 35641657548953344, 73354596206766622208, 301272202649664088951808, 2471648811030443735290891264, 40527680937730480234609755344896
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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REFERENCES
| D. G. Cantor, personal communication.
E. N. Gilbert, Enumeration of labeled graphs, Canad. J. Math., 8 (1956), 405-411.
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 518.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 7.
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).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.1.
H. S. Wilf, Generatingfunctionology, Academic Press, NY, 1990, p. 78.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..50
Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 138
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
H. S. Wilf, Generatingfunctionology, 2nd edn., Academic Press, NY, 1994, p. 87, Eq. 3.10.2.
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FORMULA
| n*2^binomial(n, 2) = Sum_k binomial(n, k)*k*a(k)*2^binomial(n-k, 2).
E.g.f.: 1+log( sum(2^binomial(n, 2)*x^n/n!, n=0..infinity) ).
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MAPLE
| t1 := 1+log( add(2^binomial(n, 2)*x^n/n!, n=0..30)): t2 := series(t1, x, 30): A001187 := n->n!*coeff(t2, x, n);
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MATHEMATICA
| g = Sum[2^Binomial[n, 2] x^n/n!, {n, 0, 20}]; Range[0, 20]! CoefficientList[Series[Log[g] + 1, {x, 0, 20}], x] (*Geoffrey Critzer, Nov 12 2011*)
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PROG
| (PARI) a(n)=n!*polcoeff(1+log(sum(k=0, n, 2^binomial(k, 2)*x^k/k!, x*O(x^n))), n)
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CROSSREFS
| Logarithmic transform of A006125 (labeled graphs). Cf. A053549.
Row sums of triangle A062734.
Sequence in context: A084284 A084285 A084286 * A093377 A178017 A131591
Adjacent sequences: A001184 A001185 A001186 * A001188 A001189 A001190
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KEYWORD
| nonn,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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