A061542 Number of connected labeled graphs with n nodes and n+3 edges.

0, 0, 0, 0, 45, 4945, 331506, 18602136, 974679363, 50088981600, 2588876118675, 136440380444544, 7389687834858186, 413138671455654144, 23901631262740105875, 1432747304604594800640, 89030607737889046580442

Offset

1,5

Links

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.

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)).

a(n) ~ 221 * n^(n+4) / 24192 * (1 - 2205*sqrt(2*Pi/n)/884). - Vaclav Kotesovec, Jan 11 2018

Mathematica

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 *)

Crossrefs

A diagonal of A343088.

Cf. A000169, A000272.

Sequence in context: A328356 A093533 A101291 * A037182 A178632 A134229

Adjacent sequences: A061539 A061540 A061541 * A061543 A061544 A061545

Keyword

easy,nice,nonn

Author

RAVELOMANANA Vlady (vlad(AT)lri.fr), May 16 2001

Status

approved