A266392 Decimal expansion of constant c in the asymptotic formula for connected labeled planar graphs on n vertices.

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

Offset

-5,1

Links

Gheorghe Coserea, Table of n, a(n) for n = -5..51003

Omer Gimenez, Marc Noy, Asymptotic enumeration and limit laws of planar graphs, J. Amer. Math. Soc. 22 (2009), 309-329.

Formula

Equals Kc(A266389), where function t->Kc(t) is defined in the PARI code.

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

Example

0.00000410436110025...

Prog

(PARI)
A266389= 0.6263716633;
Xi(t)  = (1+3*t) * (1-t)^3 / ((16*t^3));
P1(t)  = -2400 + 57952*t + 303862*t^2 + 466546*t^3;
P2(t)  = (264775 + 76679*t + 11495*t^2 + 739*t^3) * t^4;
P(t)   = P1(t) + P2(t);
Q(t)   = 400 + 1808*t + 2527*t^2 + 1155*t^3 + 237*t^4 + 17*t^5;
S(t)   = 144 + 592*t + 664*t^2 + 135*t^3 + 6*t^4 - 5*t^5;
B41(t) = log((1+t)/sqrt(1+2*t)) * (1-t)^6 * (1+3*t)^2 / (512*t^6);
B42(t) = P(t) * (1-t)^5 / (2048 * t^4 * (3+t) * Q(t));
B4(t)  = B41(t) + B42(t);
B5(t)  = -sqrt(3)/90 * (1-t)^6 / (1+t)^(3/2) * (S(t) / (t*Q(t)))^(5/2);
C5(t)  = B5(t) * (1 - 2*B4(t) / Xi(t))^(-5/2);
Kc(t)  = C5(t) / gamma(-5/2);
Kc(A266389)

Crossrefs

Cf. A096332, A266389, A266390, A266391.

Sequence in context: A122899 A223856 A153661 * A021713 A122388 A094918

Adjacent sequences: A266389 A266390 A266391 * A266393 A266394 A266395

Keyword

nonn,cons

Author

Gheorghe Coserea, Dec 29 2015

Status

approved