login
A291835
Decimal expansion of constant g in the asymptotic formula for the number of 2-connected planar graphs on n labeled nodes.
3
3, 7, 0, 4, 4, 5, 9, 4, 1, 5, 9, 4, 0, 5, 4, 8, 7, 5, 3, 5, 5, 3, 6, 3, 2, 1, 0, 1, 7, 2, 7, 3, 4, 9, 9, 2, 5, 1, 0, 5, 5, 4, 4, 1, 2, 5, 3, 0, 2, 6, 3, 3, 3, 1, 5, 1, 7, 3, 2, 3, 7, 3, 4, 3, 2, 0, 7, 5, 1, 7, 3, 8, 5, 9, 2, 0, 9
OFFSET
-5,1
LINKS
E. A. Bender, Z. Gao and N. C. Wormald, The number of labeled 2-connected planar graphs, Electron. J. Combin., 9 (2002), #R43.
FORMULA
Equals g2(A266389), where function t->g2(t) is defined in the PARI code.
Constant g where A096331(n) ~ g * n^(-7/2) * A291836^n * n!.
EXAMPLE
0.0000037044594159405487535536321017273...
PROG
(PARI)
x(t) = (1+3*t)*(1/t-1)^3/16;
y(t) = {
my(y1 = t^2 * (1-t) * (18 + 36*t + 5*t^2),
y2 = 2 * (3+t) * (1+2*t) * (1+3*t)^2);
(1+2*t)/((1+3*t) * (1-t)) * exp(-y1/y2) - 1;
};
alpha(t) = 144 + 592*t + 664*t^2 + 135*t^3 + 6*t^4 - 5*t^5;
D3(t) = {
my(d1 = 384*t^3 * (1+t)^2 * (1+2*t)^2 * (3+t)^2,
d2 = (400 + 1808*t + 2527*t^2 + 1155*t^3 + 237*t^4 + 17*t^5));
d1 * alpha(t)^(3/2) * (3*t*(1+t)*d2)^(-5/2);
};
mu(t) = {
my(mu1 = (1+t) * (3+t)^2 * (1+2*t)^2 * (1+3*t)^2 / t^3, y0 = y(t));
mu1 * y0 / ((1 + y0) * alpha(t));
};
s2(t) = {
my(y0 = y(t), a0 = alpha(t),
s20 = ((3+t) * (1+2*t) * (1+3*t))^2 / (3*t^6 * (1+t)),
s21 = 1296 + 10272*t + 30920*t^2 + 42526*t^3 + 23135*t^4,
s22 = t^5 * (1482 + 4650*t + 1358*t^2 + 405*t^3 + 30*t^4),
s23 = (1-t)*(3+t)*(1+2*t)*(1+3*t)^2 * y0 * (s21 - s22));
s20 * y0/(1+y0)^2 * (3*t^3 * (1+t)^2 * a0^2 - s23)/a0^3;
};
g2(t) = 3*x(t)^2 * D3(t)/(16*mu(t)*sqrt(Pi));
N=73; default(realprecision, N+100); t0=solve(t=.62, .63, y(t)-1);
g=g2(t0); eval(select(x->(x != "."), Vec(Str(g))[1..-101]))
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Gheorghe Coserea, Sep 03 2017
STATUS
approved