OFFSET
1,3
LINKS
Gheorghe Coserea, Table of n, a(n) for n = 1..51002
Omer Gimenez, Marc Noy, Asymptotic enumeration and limit laws of planar graphs, J. Amer. Math. Soc. 22 (2009), 309-329.
FORMULA
Equals lim E[Xn], where Xn is the number of components in a random labeled planar graph with n vertices.
Equals 1 + C0(A266389), where function t->C0(t) is defined in the PARI code.
EXAMPLE
1.0374393660275...
PROG
(PARI)
A266389= 0.6263716633;
Xi(t) = (1+3*t) * (1-t)^3 / ((16*t^3));
B01(t) = (3*t-1)^2 * (1+t)^6 * log(1+t)/(512*t^6);
B02(t) = (3*t^4 - 16*t^3 + 6*t^2 - 1) * log(1 + 3*t) / (32*t^3);
B03(t) = (1+3*t)^2 * (1-t)^6 * log(1+2*t) / (1024*t^6);
B04(t) = (1/4)*log(3+t) - (1/2)*log(t) - (3/8)*log(16);
B05(t) = (217*t^6 + 920*t^5 + 972*t^4 + 1436*t^3 + 205*t^2 - 172*t + 6);
B06(t) = (1-t)^2 / (2048 * t^4 * (1+3*t) * (3+t));
B0(t) = B01(t) - B02(t) - B03(t) + B04(t) - B05(t) * B06(t);
B21(t) = (1-t)^3 * (3*t-1) * (1+3*t) * (1+t)^3 * log(1+t) / (256*t^6);
B22(t) = (1-t)^3 * (1+3*t) * log(1+3*t) / (32*t^3);
B23(t) = (1+3*t)^2 * (1-t)^6 * log(1+2*t) / (512*t^6);
B24(t) = (1-t)^4 * (185*t^4 + 698*t^3 - 217*t^2 - 160*t + 6);
B25(t) = 1024 * t^4 * (1+3*t) * (3+t);
B2(t) = B21(t) - B22(t) + B23(t) + B24(t) / B25(t);
C0(t) = Xi(t) + B0(t) + B2(t);
1 + C0(A266389)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Gheorghe Coserea, Jan 14 2016
STATUS
approved