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A061540
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Number of connected labeled graphs with n nodes and n+1 edges.
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4
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0, 0, 0, 6, 205, 5700, 156555, 4483360, 136368414, 4432075200, 154060613850, 5720327205120, 226378594906035, 9523895202838016, 424814409531910125, 20037831121798963200, 996964614369038858060, 52198565072252054814720
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OFFSET
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1,4
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REFERENCES
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Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 407, Eq. (6.5).
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LINKS
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Sergey Serebryakov, Table of n, a(n) for n = 1..40
S. Janson, D. E. Knuth, T. Luczak and B. Pittel, The Birth of the Giant Component, Random Structures and Algorithms Vol. 4 (1993), 233-358.
S. Janson, D. E. Knuth, T. Luczak and B. Pittel, The Birth of the Giant Component, arXiv:math/9310236 [math.PR], 1993.
E. M. Wright, The Number of Connected Sparsely Edged Graphs, Journal of Graph Theory Vol. 1 (1977), 317-330.
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FORMULA
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E.g.f.: W1(x) := T(x)^4/24 * (6-T(x))/(1-T(x))^3 where T(x) is the e.g.f. for rooted labeled trees (A000169), i.e. T(x) = -LambertW(-x) = x*exp(T(x)).
a(n) ~ 5*n^(n+1)/24 * (1 - 7/5*sqrt(2*Pi/n)). - Vaclav Kotesovec, Jul 09 2013
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MAPLE
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A001864 := proc(n)
add(binomial(n, s)*s^s*(n-s)^(n-s), s=1..n-1) ;
end proc:
A061540 := proc(n)
(n-1)*(5*n^2+3*n+2)*n^(n-2)-14*A001864(n) ;
%/24 ;
end proc: # R. J. Mathar, May 10 2016 see Chapter 6.3 in Bona's Handbook of Combinatorics
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MATHEMATICA
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max = 18; t[x_] := -ProductLog[-x]; w1[x_] := t[x]^4/24*(6-t[x])/(1-t[x])^3; Drop[ CoefficientList[ Series[ w1[x], {x, 0, max}], x]*Range[0, max]!, 1] (* Jean-François Alcover, Apr 02 2012, after e.g.f. *)
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CROSSREFS
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A diagonal of A343088.
Cf. A000169, A000272, A057500, A061541, A061542, A061543, A096117, A061544, A096150, A096224.
Sequence in context: A082405 A183595 A054653 * A173370 A159307 A284768
Adjacent sequences: A061537 A061538 A061539 * A061541 A061542 A061543
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KEYWORD
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easy,nice,nonn
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AUTHOR
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RAVELOMANANA Vlady (vlad(AT)lri.fr), May 16 2001
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STATUS
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approved
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