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%I A291835 #20 Sep 05 2017 03:31:30
%S A291835 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,
%T A291835 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,
%U A291835 1,7,3,8,5,9,2,0,9
%N A291835 Decimal expansion of constant g in the asymptotic formula for the number of 2-connected planar graphs on n labeled nodes.
%H A291835 Gheorghe Coserea, <a href="/A291835/b291835.txt">Table of n, a(n) for n = -5..54998</a>
%H A291835 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 A291835 Equals g2(A266389), where function t->g2(t) is defined in the PARI code.
%F A291835 Constant g where A096331(n) ~  g * n^(-7/2) * A291836^n * n!.
%e A291835 0.0000037044594159405487535536321017273...
%o A291835 (PARI)
%o A291835 x(t)     = (1+3*t)*(1/t-1)^3/16;
%o A291835 y(t)     = {
%o A291835   my(y1  = t^2 * (1-t) * (18 + 36*t + 5*t^2),
%o A291835      y2  = 2 * (3+t) * (1+2*t) * (1+3*t)^2);
%o A291835   (1+2*t)/((1+3*t) * (1-t)) * exp(-y1/y2) - 1;
%o A291835 };
%o A291835 alpha(t) = 144 + 592*t + 664*t^2 + 135*t^3 + 6*t^4 - 5*t^5;
%o A291835 D3(t)    = {
%o A291835   my(d1  = 384*t^3 * (1+t)^2 * (1+2*t)^2 * (3+t)^2,
%o A291835      d2  = (400 + 1808*t + 2527*t^2 + 1155*t^3 + 237*t^4 + 17*t^5));
%o A291835   d1 * alpha(t)^(3/2) * (3*t*(1+t)*d2)^(-5/2);
%o A291835 };
%o A291835 mu(t)    = {
%o A291835   my(mu1 = (1+t) * (3+t)^2 * (1+2*t)^2 * (1+3*t)^2 / t^3, y0 = y(t));
%o A291835   mu1 * y0 / ((1 + y0) * alpha(t));
%o A291835 };
%o A291835 s2(t)    = {
%o A291835   my(y0  = y(t), a0 = alpha(t),
%o A291835      s20 = ((3+t) * (1+2*t) * (1+3*t))^2 / (3*t^6 * (1+t)),
%o A291835      s21 = 1296 + 10272*t + 30920*t^2 + 42526*t^3 + 23135*t^4,
%o A291835      s22 = t^5 * (1482 + 4650*t + 1358*t^2 + 405*t^3 + 30*t^4),
%o A291835      s23 = (1-t)*(3+t)*(1+2*t)*(1+3*t)^2 * y0 * (s21 - s22));
%o A291835   s20 * y0/(1+y0)^2 * (3*t^3 * (1+t)^2 * a0^2 - s23)/a0^3;
%o A291835 };
%o A291835 g2(t)    = 3*x(t)^2 * D3(t)/(16*mu(t)*sqrt(Pi));
%o A291835 N=73; default(realprecision, N+100); t0=solve(t=.62, .63, y(t)-1);
%o A291835 g=g2(t0); eval(select(x->(x != "."), Vec(Str(g))[1..-101]))
%Y A291835 Cf. A096331, A266389, A291836.
%K A291835 nonn,cons
%O A291835 -5,1
%A A291835 _Gheorghe Coserea_, Sep 03 2017

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