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A212809 Decimal expansion of radius of convergence of g.f. for unlabeled trees (A000055).

%I A212809 #27 Dec 12 2024 17:36:02
%S A212809 3,3,8,3,2,1,8,5,6,8,9,9,2,0,7,6,9,5,1,9,6,1,1,2,6,2,5,7,1,7,0,1,7,0,
%T A212809 5,3,1,8,3,7,7,4,6,0,7,5,3,2,9,6,7,7,9,5,5,7,2,3,0,3,7,7,6,2,5,7,6,6,
%U A212809 6,0,5,0,1,8,9,6,2,0,7,6,6,5,6,3,5,2,8,7,9,8,3,6,7,3
%N A212809 Decimal expansion of radius of convergence of g.f. for unlabeled trees (A000055).
%D A212809 Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.6, p. 296.
%H A212809 Michael Drmota and Bernhard Gittenberger, <a href="https://doi.org/10.1016/j.ejc.2010.05.011">The shape of unlabeled rooted random trees</a>, Eur. J. Comb. 31 (2010) no 8, 2028-2063.
%H A212809 E. M. Palmer and A. J. Schwenk, <a href="http://dx.doi.org/10.1016/0095-8956(79)90073-X">On the number of trees in a random forest</a>, J. Combin. Theory, B 27 (1979), 109-121.
%F A212809 Equals 1/A051491. - _Vaclav Kotesovec_, Jul 29 2013
%e A212809 0.338321856899208...
%t A212809 digits = 95; max = 200;
%t A212809 s[n_, k_] := s[n, k] = a[n + 1 - k] + If[n < 2*k, 0, s[n - k, k]];
%t A212809 a[1] = 1;
%t A212809 a[n_] := a[n] = Sum[a[k]*s[n - 1, k]*k, {k, 1, n - 1}]/(n - 1);
%t A212809 A[x_] := Sum[a[k]*x^k, {k, 0, max}];
%t A212809 eq = Log[c] == 1 + Sum[A[c^-k]/k, {k, 2, max}];
%t A212809 r = 1/c /. FindRoot[eq, {c, 3}, WorkingPrecision -> digits + 5];
%t A212809 RealDigits[r, 10, digits] // First (* _Jean-François Alcover_, Aug 10 2016 *)
%Y A212809 Cf. A000055.
%K A212809 nonn,cons
%O A212809 0,1
%A A212809 _N. J. A. Sloane_, May 29 2012
%E A212809 More terms from _Vaclav Kotesovec_, Jul 29 2013