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A061542
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Number of connected labeled graphs with n nodes and n+3 edges.
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5
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0, 0, 0, 0, 45, 4945, 331506, 18602136, 974679363, 50088981600, 2588876118675, 136440380444544, 7389687834858186, 413138671455654144, 23901631262740105875, 1432747304604594800640, 89030607737889046580442
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OFFSET
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1,5
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LINKS
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Sergey Serebryakov, Table of n, a(n) for n = 1..100
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.: W2(x) = 1/5760*T(x)^5*(2160 + 9320*T(x) - 12576*T(x)^2 + 9864*T(x)^3 - 4081*T(x)^4 + 914*T(x)^5 - 76*T(x)^6)/((1 - T(x))^9), 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) ~ 221 * n^(n+4) / 24192 * (1 - 2205*sqrt(2*Pi/n)/884). - Vaclav Kotesovec, Jan 11 2018
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MATHEMATICA
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terms = 17; T[x_] = -ProductLog[-x];
W2[x_] = (1/5760)*T[x]^5*((2160 + 9320*T[x] - 12576*T[x]^2 + 9864*T[x]^3 - 4081*T[x]^4 + 914*T[x]^5 - 76*T[x]^6)/(1 - T[x])^9) + O[x]^(terms+1);
Drop[CoefficientList[W2[x], x]*Range[0, terms]!, 1](* Jean-François Alcover, Nov 04 2011, updated Jan 11 2018 *)
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CROSSREFS
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A diagonal of A343088.
Cf. A000169, A000272.
Sequence in context: A328356 A093533 A101291 * A037182 A178632 A134229
Adjacent sequences: A061539 A061540 A061541 * A061543 A061544 A061545
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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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