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A001187 Number of connected labeled graphs with n nodes.
(Formerly M3671 N1496)
24
1, 1, 1, 4, 38, 728, 26704, 1866256, 251548592, 66296291072, 34496488594816, 35641657548953344, 73354596206766622208, 301272202649664088951808, 2471648811030443735290891264, 40527680937730480234609755344896 (list; graph; refs; listen; history; internal format)
OFFSET

0,4

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.

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.

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) ).

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);

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*)

PROG

(PARI) a(n)=n!*polcoeff(1+log(sum(k=0, n, 2^binomial(k, 2)*x^k/k!, x*O(x^n))), n)

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

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane (njas(AT)research.att.com).

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Last modified February 17 18:27 EST 2012. Contains 206066 sequences.