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%I #38 Nov 02 2023 04:22:27
%S 0,0,0,1,120,6165,258125,10230360,405918324,16530124800,699126562530,
%T 30884683104000,1428626760992860,69248819808744576,
%U 3516693960681822375,186964957159176734720,10395215954531344335000,603712553730550509035520,36575888366817680447745924
%N Number of connected labeled graphs with n nodes and n+2 edges.
%H Vaclav Kotesovec, <a href="/A061541/b061541.txt">Table of n, a(n) for n = 1..380</a> (first 100 terms from T. D. Noe)
%H S. Janson, D. E. Knuth, T. Luczak, and B. Pittel, <a href="https://dx.doi.org/10.1002/rsa.3240040303">The Birth of the Giant Component</a>, Random Structures and Algorithms Vol. 4 (1993), 233-358.
%H S. Janson, D. E. Knuth, T. Luczak, and B. Pittel, <a href="https://arxiv.org/abs/math/9310236">The Birth of the Giant Component</a>, arXiv:math/9310236 [math.PR], 1993.
%H E. M. Wright, <a href="https://dx.doi.org/10.1002/jgt.3190010407">The Number of Connected Sparsely Edged Graphs</a>, Journal of Graph Theory Vol. 1 (1977), 317-330.
%F 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)).
%F a(n) ~ 5*n^(n+5/2)*sqrt(2*Pi)/256 * (1 - 56*sqrt(2)/(9*sqrt(Pi*n))). - _Vaclav Kotesovec_, Apr 06 2014
%t 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 *)
%o (PARI) N=66; x='x+O('x^N); /* that many terms */
%o T=sum(n=1,N,n^(n-1)/n!*x^n); /* e.g.f. of A000169 */
%o egf=1/48*T^4*(2+28*T-23*T^2+9*T^3-T^4)/(1-T)^6;
%o Vec(serlaplace(egf)) /* show terms, starting with 1 */
%o /* _Joerg Arndt_, Jul 11 2011 */
%Y A diagonal of A343088.
%Y Cf. A000169, A000272, A057500, A061541, A061542, etc.
%K easy,nice,nonn
%O 1,5
%A RAVELOMANANA Vlady (vlad(AT)lri.fr), May 16 2001
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