A061541 Number of connected labeled graphs with n nodes and n+2 edges.

0, 0, 0, 1, 120, 6165, 258125, 10230360, 405918324, 16530124800, 699126562530, 30884683104000, 1428626760992860, 69248819808744576, 3516693960681822375, 186964957159176734720, 10395215954531344335000, 603712553730550509035520, 36575888366817680447745924

Offset: 1

Author

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

Keywords

• easy
• nice
• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..380 (first 100 terms from T. D. Noe)
• Steven R. Finch, An exceptional convolutional recurrence, arXiv:2408.12440 [math.CO], 22 Aug 2024.
• S. Janson, D. E. Knuth, T. Łuczak, and B. Pittel, The Birth of the Giant Component, Random Structures and Algorithms Vol. 4 (1993), 233-358.
• S. Janson, D. E. Knuth, T. Łuczak, 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.

Crossrefs

A diagonal of A343088.

Cf. A000169, A000272, A057500, A061541, A061542, etc.

Programs

Mathematica

f[x_] = (1/(48*(1 + ProductLog[-x])^6))* ProductLog[-x]^4*(2 - 28*ProductLog[-x] - 23*ProductLog[-x]^2 - 9*ProductLog[-x]^3 - ProductLog[-x]^4); Rest[CoefficientList[Series[f[x], {x, 0, 17}], x]*Range[0, 17]!] (* Jean-François Alcover, Jul 11 2011, after formula *)

PARI

N=66; x='x+O('x^N); /* that many terms */
T=sum(n=1,N,n^(n-1)/n!*x^n); /* e.g.f. of A000169 */
egf=1/48*T^4*(2+28*T-23*T^2+9*T^3-T^4)/(1-T)^6;
Vec(serlaplace(egf)) /* show terms, starting with 1 */
/* Joerg Arndt, Jul 11 2011 */

Formula

E.g.f.: W2(x) = (1/48)*T(x)^4*(2 + 28*T(x) - 23*T(x)^2 + 9*T(x)^3 - T(x)^4)/(1 - T(x))^6, 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+5/2)*sqrt(2*Pi)/256 * (1 - 56*sqrt(2)/(9*sqrt(Pi*n))). - Vaclav Kotesovec, Apr 06 2014