A240943 Decimal expansion of the radius of convergence of Wedderburn-Etherington numbers g.f.

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, 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, 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

Offset: 0

Author

Jean-François Alcover, Aug 04 2014

Keywords

• nonn
• cons

Examples

			0.4026975036714412909690453486510838034175567216249726592910534646...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.6 Otter's Tree Enumeration Constants, p. 297.

Links

• Nils Berglund, Yvain Bruned, BPHZ renormalisation and vanishing subcriticality limit of the fractional Phi_d^3 model, arXiv:1907.13028 [math.PR], 2019.
• Nils Berglund, Christian Kuehn, Model Spaces of Regularity Structures for Space-Fractional SPDEs, Journal of Statistical Physics, Springer Verlag, 2017, 168 (2), pp.331-368; HAL Id : hal-01432157.
• Nicolas Broutin and Philippe Flajolet, The height of random binary unlabelled trees, arXiv:0807.2365 [math.CO], 2008.
• Eric Weisstein's World of Mathematics, Weakly binary tree

Crossrefs

Cf. A001190, A086317.

Programs

Mathematica

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
(* 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 *)

Formula

1/A086317.