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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 (list; constant; graph; refs; listen; history; text; internal format)
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

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Last modified February 23 08:00 EST 2019. Contains 320420 sequences. (Running on oeis4.)