A291838 - OEIS

%I #12 Sep 05 2017 04:09:24

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

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

%U 3,9,9,0,5,8,9,2,9,1,6

%N Decimal expansion of the constant factor k in the asymptotic formula for A291837.

%H Gheorghe Coserea, <a href="/A291838/b291838.txt">Table of n, a(n) for n = -5..54998</a>

%H E. A. Bender, Z. Gao and N. C. Wormald, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v9i1r43">The number of labeled 2-connected planar graphs</a>, Electron. J. Combin., 9 (2002), #R43.

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

%F Constant k where A291837(n) ~ k * n^(-4) * A291836^n * n! (see Bender link).

%e 0.000002389700772064025828320154831485...

%o (PARI)

%o x(t)     = (1+3*t)*(1/t-1)^3/16;

%o y(t)     = {

%o   my(y1  = t^2 * (1-t) * (18 + 36*t + 5*t^2),

%o      y2  = 2 * (3+t) * (1+2*t) * (1+3*t)^2);

%o   (1+2*t)/((1+3*t) * (1-t)) * exp(-y1/y2) - 1;

%o };

%o alpha(t) = 144 + 592*t + 664*t^2 + 135*t^3 + 6*t^4 - 5*t^5;

%o D3(t)    = {

%o   my(d1  = 384*t^3 * (1+t)^2 * (1+2*t)^2 * (3+t)^2,

%o      d2  = (400 + 1808*t + 2527*t^2 + 1155*t^3 + 237*t^4 + 17*t^5));

%o   d1 * alpha(t)^(3/2) * (3*t*(1+t)*d2)^(-5/2);

%o };

%o mu(t)    = {

%o   my(mu1 = (1+t) * (3+t)^2 * (1+2*t)^2 * (1+3*t)^2 / t^3, y0 = y(t));

%o   mu1 * y0 / ((1 + y0) * alpha(t));

%o };

%o s2(t)    = {

%o   my(y0  = y(t), a0 = alpha(t),

%o      s20 = ((3+t) * (1+2*t) * (1+3*t))^2 / (3*t^6 * (1+t)),

%o      s21 = 1296 + 10272*t + 30920*t^2 + 42526*t^3 + 23135*t^4,

%o      s22 = t^5 * (1482 + 4650*t + 1358*t^2 + 405*t^3 + 30*t^4),

%o      s23 = (1-t)*(3+t)*(1+2*t)*(1+3*t)^2 * y0 * (s21 - s22));

%o   s20 * y0/(1+y0)^2 * (3*t^3 * (1+t)^2 * a0^2 - s23)/a0^3;

%o };

%o K(t) = 3*x(t)^2*D3(t)/(16*mu(t)*Pi*sqrt(2*s2(t)));

%o N=75; default(realprecision, N+100); t0 = solve(t=.62, .63, y(t)-1);

%o k=K(t0); eval(select(x->(x != "."), Vec(Str(k))[1..-101]))

%Y Cf. A266389, A291837.

%K nonn,cons

%O -5,1

%A _Gheorghe Coserea_, Sep 05 2017