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Decimal expansion of constant g in the asymptotic formula for labeled planar graphs on n vertices.
5

%I #39 Feb 23 2016 13:50:14

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

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

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

%N Decimal expansion of constant g in the asymptotic formula for labeled planar graphs on n vertices.

%H Gheorghe Coserea, <a href="/A266391/b266391.txt">Table of n, a(n) for n = -5..50993</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 K(A266389), where function t->K(t) is defined in the PARI code.

%F Constant g where A066537(n) ~ g * A266390^n * n^(-7/2) * n!.

%e 0.000004260938569161439...

%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 P1(t) = -2400 + 57952*t + 303862*t^2 + 466546*t^3;

%o P2(t) = (264775 + 76679*t + 11495*t^2 + 739*t^3) * t^4;

%o P(t) = P1(t) + P2(t);

%o Q(t) = 400 + 1808*t + 2527*t^2 + 1155*t^3 + 237*t^4 + 17*t^5;

%o S(t) = 144 + 592*t + 664*t^2 + 135*t^3 + 6*t^4 - 5*t^5;

%o B41(t) = log((1+t)/sqrt(1+2*t)) * (1-t)^6 * (1+3*t)^2 / (512*t^6);

%o B42(t) = P(t) * (1-t)^5 / (2048 * t^4 * (3+t) * Q(t));

%o B4(t) = B41(t) + B42(t);

%o B5(t) = -sqrt(3)/90 * (1-t)^6 / (1+t)^(3/2) * (S(t) / (t*Q(t)))^(5/2);

%o C0(t) = Xi(t) + B0(t) + B2(t);

%o C5(t) = B5(t) * (1 - 2*B4(t) / Xi(t))^(-5/2);

%o K(t) = exp(C0(t)) * C5(t) / gamma(-5/2);

%o K(A266389)

%Y Cf. A066537, A266389, A266390.

%K nonn,cons

%O -5,1

%A _Gheorghe Coserea_, Dec 28 2015