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

0,1

References

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

Links

Table of n, a(n) for n=0..101.

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

Formula

1/A086317.

Example

0.4026975036714412909690453486510838034175567216249726592910534646...

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 *)

Crossrefs

Cf. A001190, A086317.

Sequence in context: A322259 A057607 A377644 * A271823 A011352 A275983

Adjacent sequences: A240940 A240941 A240942 * A240944 A240945 A240946

Keyword

nonn,cons,changed

Author

Jean-François Alcover, Aug 04 2014

Status

approved