A267411 - OEIS

%I #7 Jan 18 2016 21:51:08

%S 3,9,0,5,1,8,0,2,8,2,4,5,9,1,1,1,4,7,5,5,8,5,0,6,2,6,2,1,7,3,0,9,5,0,

%T 6,7,0,4,6,4,1,1,3,0,7,6,5,2,6,0,2,9,3,5,2,1,9,0,0,6,1,9,4,5,7,1,5,5,

%U 1,4,1,5,3,5,6,1,3,6,3,1,4,2,3,9

%N Decimal expansion of the constant describing the mean number of 2-connected components in a random connected labeled planar graph on n vertices.

%H Gheorghe Coserea, <a href="/A267411/b267411.txt">Table of n, a(n) for n = -1..51000</a>

%H Omer Gimenez, Marc Noy, <a href="http://dx.doi.org/10.1090/S0894-0347-08-00624-3">Asymptotic enumeration and limit laws of planar graphs</a>, J. Amer. Math. Soc. 22 (2009), 309-329.

%F Equals lim E[Xn]/n, where Xn is the number of 2-connected components in a random connected labeled planar graph with n vertices; also equals lim Var(Xn)/n.

%F Equals Kz(A266389), where function t->Kz(t) is defined in the PARI code.

%e 0.039051802824591114...

%o (PARI)

%o A266389= 0.6263716633;

%o Xi(t)  = (1+3*t) * (1-t)^3 / ((16*t^3));

%o A1(t)  = log(1+t) * (3*t-1) * (1+t)^3 / (16*t^3);

%o A2(t)  = log(1+2*t) * (1+3*t) * (1-t)^3 / (32*t^3);

%o A3(t)  = (1-t) * (185*t^4 + 698*t^3 - 217*t^2 - 160*t + 6);

%o A4(t)  = 64*t * (1+3*t)^2 * (3+t);

%o A(t)   = A1(t) + A2(t) + A3(t) / A4(t);

%o R(t)   = 1/16 * sqrt(1+3*t) * (1/t - 1)^3 * exp(A(t));

%o Kz(t)  = log(Xi(t)/R(t));

%o Kz(A266389)

%Y Cf. A266389, A266392.

%K nonn,cons

%O -1,1

%A _Gheorghe Coserea_, Jan 14 2016