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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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6
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0, 0, 0, 0, 45, 4945, 331506, 18602136, 974679363, 50088981600, 2588876118675, 136440380444544, 7389687834858186, 413138671455654144, 23901631262740105875, 1432747304604594800640, 89030607737889046580442
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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REFERENCES
| 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.
E. M. Wright, The Number of Connected Sparsely Edged Graphs, Journal of Graph Theory Vol. 1 (1977), 317-330.
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FORMULA
| 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)).
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MATHEMATICA
| max = 17; w[x_] = -ProductLog[-x]; w2[x_] = Sum[c[k]*(x^k/k!), {k, 0, max}]; se = Series[ w2[x] - (1/5760)* w[x]^5*((2160 + 9320*w[x] - 12576*w[x]^2 + 9864*w[x]^3 -4081*w[x]^4 + 914*w[x]^5 - 76*w[x]^6)/(1 - w[x])^9), {x, 0, max}]; coes = CoefficientList[se, x]; sol = First[ Solve[ Thread[ coes == 0]]]; a[n_] := c[n] /. sol; Table[a[n], {n, 1, max}](* From Jean-François Alcover, Nov 04 2011 *)
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CROSSREFS
| Cf. A000169, A000272.
Sequence in context: A036521 A093533 A101291 * A037182 A178632 A134229
Adjacent sequences: A061539 A061540 A061541 * A061543 A061544 A061545
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KEYWORD
| easy,nice,nonn
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AUTHOR
| RAVELOMANANA Vlady (vlad(AT)lri.fr), May 16 2001
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