A291835 Decimal expansion of constant g in the asymptotic formula for the number of 2-connected planar graphs on n labeled nodes.

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

Gheorghe Coserea, Table of n, a(n) for n = -5..54998

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

Cf. A096331, A266389, A291836.

Sequence in context: A005600 A180873 A021031 * A197835 A197005 A199778

Adjacent sequences: A291832 A291833 A291834 * A291836 A291837 A291838

Keyword

nonn,cons

Author

Gheorghe Coserea, Sep 03 2017

Status

approved