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Decimal expansion of the constant describing the average number of edges of a random labeled planar graph with n vertices.
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%I #13 Jan 18 2016 21:50:28

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

%T 6,6,3,8,6,5,1,9,5,6,2,5,1,1,5,5,4,2,1,5,9,2,7,9,7,1,8,2,7,1,7,7,1,9,

%U 5,9,7,6,4,8,7,0,3,8,8,5,0,8,3,2

%N Decimal expansion of the constant describing the average number of edges of a random labeled planar graph with n vertices.

%H Gheorghe Coserea, <a href="/A267409/b267409.txt">Table of n, a(n) for n = 1..51004</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 edges of a random labeled planar graph with n vertices.

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

%e 2.21326523857442...

%o (PARI)

%o A266389= 0.6263716633;

%o Y1(t) = t^2 * (1-t) * (18 + 36*t + 5*t^2);

%o Y2(t) = 2 * (3+t) * (1+2*t) * (1+3*t)^2;

%o Y(t) = (1+2*t) / ((1+3*t)*(1-t)) * exp(-Y1(t)/Y2(t)) - 1;

%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 Km(t) = -R'(t)/(R(t)*Y'(t));

%o Km(A266389)

%Y Cf. A266389, A266390, A267410, A267412.

%K nonn,cons

%O 1,1

%A _Gheorghe Coserea_, Jan 13 2016