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A266391 Decimal expansion of constant g in the asymptotic formula for labeled planar graphs on n vertices. 5
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, 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, 2, 5, 8, 4, 8, 2, 0, 7, 6, 8, 5, 0, 0, 4, 9, 0 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

-5,1

LINKS

Gheorghe Coserea, Table of n, a(n) for n = -5..50993

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

FORMULA

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

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

EXAMPLE

0.000004260938569161439...

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);

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);

C0(t)  = Xi(t) + B0(t) + B2(t);

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

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

K(A266389)

CROSSREFS

Cf. A066537, A266389, A266390.

Sequence in context: A141674 A329323 A178394 * A091664 A010317 A131749

Adjacent sequences:  A266388 A266389 A266390 * A266392 A266393 A266394

KEYWORD

nonn,cons

AUTHOR

Gheorghe Coserea, Dec 28 2015

STATUS

approved

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Last modified May 17 19:46 EDT 2021. Contains 343988 sequences. (Running on oeis4.)