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%I #20 Sep 05 2017 03:31:30
%S 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 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 1,7,3,8,5,9,2,0,9
%N Decimal expansion of constant g in the asymptotic formula for the number of 2-connected planar graphs on n labeled nodes.
%H Gheorghe Coserea, <a href="/A291835/b291835.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 g2(A266389), where function t->g2(t) is defined in the PARI code.
%F Constant g where A096331(n) ~ g * n^(-7/2) * A291836^n * n!.
%e 0.0000037044594159405487535536321017273...
%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 g2(t) = 3*x(t)^2 * D3(t)/(16*mu(t)*sqrt(Pi));
%o N=73; default(realprecision, N+100); t0=solve(t=.62, .63, y(t)-1);
%o g=g2(t0); eval(select(x->(x != "."), Vec(Str(g))[1..-101]))
%Y Cf. A096331, A266389, A291836.
%K nonn,cons
%O -5,1
%A _Gheorghe Coserea_, Sep 03 2017
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