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%I #8 Jan 18 2016 21:50:56
%S 1,0,3,7,4,3,9,3,6,6,0,2,7,5,0,6,6,1,4,8,7,3,9,0,2,0,6,5,5,9,8,7,3,1,
%T 5,0,1,4,0,3,2,2,5,9,6,0,2,4,6,3,2,0,1,2,8,3,9,3,5,6,3,2,2,7,8,0,0,3,
%U 0,6,7,5,8,7,6,1,3,8,7,5,1,0,0,7
%N Decimal expansion of the constant describing the expected number of components in a random labeled planar graph on n vertices.
%H Gheorghe Coserea, <a href="/A267412/b267412.txt">Table of n, a(n) for n = 1..51002</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], where Xn is the number of components in a random labeled planar graph with n vertices.
%F Equals 1 + C0(A266389), where function t->C0(t) is defined in the PARI code.
%e 1.0374393660275...
%o (PARI)
%o A266389= 0.6263716633;
%o Xi(t) = (1+3*t) * (1-t)^3 / ((16*t^3));
%o B01(t) = (3*t-1)^2 * (1+t)^6 * log(1+t)/(512*t^6);
%o B02(t) = (3*t^4 - 16*t^3 + 6*t^2 - 1) * log(1 + 3*t) / (32*t^3);
%o B03(t) = (1+3*t)^2 * (1-t)^6 * log(1+2*t) / (1024*t^6);
%o B04(t) = (1/4)*log(3+t) - (1/2)*log(t) - (3/8)*log(16);
%o B05(t) = (217*t^6 + 920*t^5 + 972*t^4 + 1436*t^3 + 205*t^2 - 172*t + 6);
%o B06(t) = (1-t)^2 / (2048 * t^4 * (1+3*t) * (3+t));
%o B0(t) = B01(t) - B02(t) - B03(t) + B04(t) - B05(t) * B06(t);
%o B21(t) = (1-t)^3 * (3*t-1) * (1+3*t) * (1+t)^3 * log(1+t) / (256*t^6);
%o B22(t) = (1-t)^3 * (1+3*t) * log(1+3*t) / (32*t^3);
%o B23(t) = (1+3*t)^2 * (1-t)^6 * log(1+2*t) / (512*t^6);
%o B24(t) = (1-t)^4 * (185*t^4 + 698*t^3 - 217*t^2 - 160*t + 6);
%o B25(t) = 1024 * t^4 * (1+3*t) * (3+t);
%o B2(t) = B21(t) - B22(t) + B23(t) + B24(t) / B25(t);
%o C0(t) = Xi(t) + B0(t) + B2(t);
%o 1 + C0(A266389)
%Y Cf. A266389, A266390, A267409, A267410.
%K nonn,cons
%O 1,3
%A _Gheorghe Coserea_, Jan 14 2016
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