A267412 Decimal expansion of the constant describing the expected number of components in a random labeled planar graph on n vertices.

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, 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, 0, 6, 7, 5, 8, 7, 6, 1, 3, 8, 7, 5, 1, 0, 0, 7

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

Cf. A266389, A266390, A267409, A267410.

Sequence in context: A341605 A256676 A349010 * A087941 A278389 A021271

Adjacent sequences: A267409 A267410 A267411 * A267413 A267414 A267415

Keyword

nonn,cons

Author

Gheorghe Coserea, Jan 14 2016

Status

approved