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Decimal expansion of constant c in the asymptotic formula for connected labeled planar graphs on n vertices.
4

%I #22 Feb 23 2016 13:50:31

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

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

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

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

%H Gheorghe Coserea, <a href="/A266392/b266392.txt">Table of n, a(n) for n = -5..51003</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 Kc(A266389), where function t->Kc(t) is defined in the PARI code.

%F Constant c where A096332(n) ~ c * A266390^n * n^(-7/2) * n!.

%e 0.00000410436110025...

%o (PARI)

%o A266389= 0.6263716633;

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

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

%o Kc(t) = C5(t) / gamma(-5/2);

%o Kc(A266389)

%Y Cf. A096332, A266389, A266390, A266391.

%K nonn,cons

%O -5,1

%A _Gheorghe Coserea_, Dec 29 2015