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A261340 Decimal expansion of the radius of convergence of the generating function of A000598, the number of rooted ternary trees of n vertices.

%I A261340 #24 Sep 08 2019 07:21:21
%S A261340 3,5,5,1,8,1,7,4,2,3,1,4,3,7,7,3,9,2,8,8,2,2,4,4,4,7,3,6,4,7,6,3,2,6,
%T A261340 3,6,7,0,8,7,4,6,9,5,4,1,7,5,3,2,2,1,3,4,2,3,8,1,2,9,4,9,9,7,1,2,8,0,
%U A261340 0,1,8,0,5,7,5,5,5,7,8,2,8,8,6,7,9,8,1,3,8,1,0,8,2,4,1,6,7
%N A261340 Decimal expansion of the radius of convergence of the generating function of A000598, the number of rooted ternary trees of n vertices.
%D A261340 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.6 Otter's tree enumeration constants, p. 298.
%H A261340 Philippe Flajolet and Robert Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; p. 478.
%e A261340 0.35518174231437739288224447364763263670874695417532...
%t A261340 digits = 97; m = 2 digits + 10; For[gf = 0; i = 0, i <= m, i++, gf = Series[1 + x*(gf^3/6 + (gf /. x -> x^2)*gf/2 + (gf /. x -> x^3)/3), {x, 0, m + 1}] // Normal];
%t A261340 g[r_] := Module[{r2, r3, X, ym}, r2 = gf /. x -> r^2; r3 = gf /. x -> r^3; X[y_] = (y - 1)/(y^3/6 + r2*y/2 + r3/3); ym = y /. FindRoot[X'[y] == 0, {y, 2}, WorkingPrecision -> digits + 5]; X[ym]]; rho = FixedPoint[g, 1/3, SameTest -> (Abs[#1 - #2] < 10^-digits &)]; RealDigits[rho, 10, digits] // First
%Y A261340 Cf. A000598, A000642, A002094, A121331, A244399.
%K A261340 nonn,cons
%O A261340 0,1
%A A261340 _Jean-François Alcover_, Aug 15 2015
%E A261340 More digits from _Vaclav Kotesovec_, Aug 15 2015
%E A261340 More digits and Mma code updated by _Jean-François Alcover_, Apr 18 2016