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%I #22 Oct 17 2019 01:59:56
%S 4,0,2,6,9,7,5,0,3,6,7,1,4,4,1,2,9,0,9,6,9,0,4,5,3,4,8,6,5,1,0,8,3,8,
%T 0,3,4,1,7,5,5,6,7,2,1,6,2,4,9,7,2,6,5,9,2,9,1,0,5,3,4,6,4,6,0,7,6,4,
%U 2,7,2,8,9,6,6,5,2,4,2,5,8,4,1,6,4,1,6,0,9,6,0,2,6,2,1,7,2,0,5,9,5,2
%N Decimal expansion of the radius of convergence of Wedderburn-Etherington numbers g.f.
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.6 Otter's Tree Enumeration Constants, p. 297.
%H Nils Berglund, Yvain Bruned, <a href="https://arxiv.org/abs/1907.13028">BPHZ renormalisation and vanishing subcriticality limit of the fractional Phi_d^3 model</a>, arXiv:1907.13028 [math.PR], 2019.
%H Nils Berglund, Christian Kuehn, <a href="https://hal.archives-ouvertes.fr/hal-01432157">Model Spaces of Regularity Structures for Space-Fractional SPDEs</a>, Journal of Statistical Physics, Springer Verlag, 2017, 168 (2), pp.331-368; HAL Id : hal-01432157.
%H Nicolas Broutin and Philippe Flajolet, <a href="http://arxiv.org/abs/0807.2365">The height of random binary unlabelled trees</a>, arXiv:0807.2365 [math.CO], 2008.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WeaklyBinaryTree.html">Weakly binary tree</a>
%F 1/A086317.
%e 0.4026975036714412909690453486510838034175567216249726592910534646...
%t digits = 102; n0 = 50; dn = 50; Clear[rho]; rho[n_] := rho[n] = (Clear[c]; c[0] = 0; y[z_] = Sum[c[k]*z^k, {k, 0, n}]; eq[0] = Rest[ Thread[CoefficientList[(-2*z + 2*y[z] - y[z]^2 - y[z^2])/2, z] == 0]]; s[1] = First[Solve[First[eq[0]], c[1]]]; Do[eq[k-1] = Rest[eq[k-2]] /. s[k-1]; s[k] = First[Solve[First[eq[k-1]], c[k]]], {k, 2, n}]; z /. FindRoot[ 2*z + y[z^2] == 1 /. Flatten[Table[s[k], {k, 1, n}]], {z, 1/2}, WorkingPrecision -> digits+10]); rho[n0]; rho[n = n0 + dn]; While[RealDigits[rho[n], 10, digits] != RealDigits[rho[n - dn], 10, digits], Print["n = ", n]; n = n + dn]; RealDigits[rho[n], 10, digits] // First
%t (* or, after A086317: *) Clear[c, xi]; c[0] = 2; c[n_] := c[n] = c[n-1]^2 + 2; xi[n_Integer] := xi[n] = c[n]^(2^-n); xi[5]; xi[n = 10]; While[RealDigits[xi[n], 10, digits] != RealDigits[xi[n-5], 10, digits], n = n+5]; RealDigits[1/xi[n], 10, digits] // First (* _Jean-François Alcover_, Aug 04 2014 *)
%Y Cf. A001190, A086317.
%K nonn,cons
%O 0,1
%A _Jean-François Alcover_, Aug 04 2014
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