A187770 Decimal expansion of Otter's asymptotic constant beta for the number of rooted trees.

4, 3, 9, 9, 2, 4, 0, 1, 2, 5, 7, 1, 0, 2, 5, 3, 0, 4, 0, 4, 0, 9, 0, 3, 3, 9, 1, 4, 3, 4, 5, 4, 4, 7, 6, 4, 7, 9, 8, 0, 8, 5, 4, 0, 7, 9, 4, 0, 1, 1, 9, 8, 5, 7, 6, 5, 3, 4, 9, 3, 5, 4, 5, 0, 2, 2, 6, 3, 5, 4, 0, 0, 4, 2, 0, 4, 7, 6, 4, 6, 0, 5, 3, 7, 9, 8, 6

Offset

0,1

Comments

A000081(n) ~ 0.439924012571 * alpha^n * n^(-3/2), alpha = 2.95576528565199497... (see A051491)

References

S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 5.6., p.296

D. E. Knuth, Fundamental Algorithms, 3d Ed. 1997, p. 396.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..1799, (this constant was computed by David Broadhurst in November 1999)

Amirmohammad Farzaneh, Mihai-Alin Badiu, and Justin P. Coon, On Random Tree Structures, Their Entropy, and Compression, arXiv:2309.09779 [cs.IT], 2023.

Eric Weisstein's World of Mathematics, Rooted Tree

Example

0.43992401257102530404090339143454476479808540794...

Mathematica

digits = 87; max = 250; s[n_, k_] := s[n, k] = a[n+1-k] + If[n < 2*k, 0, s[n-k, k]]; a[1] = 1; a[n_] := a[n] = Sum[a[k]*s[n-1, k]*k, {k, 1, n-1}]/(n-1); A[x_] := Sum[a[k]*x^k, {k, 0, max}]; APrime[x_] := Sum[k*a[k]*x^(k-1), {k, 0, max}]; eq = Log[c] == 1 + Sum[A[c^(-k)]/k, {k, 2, max}]; alpha = c /. FindRoot[eq, {c, 3}, WorkingPrecision -> digits+5]; b = Sqrt[(1 + Sum[APrime[alpha^-k]/alpha^k, {k, 2, max}])/(2*Pi)]; RealDigits[b, 10, digits] // First (* Jean-François Alcover, Sep 24 2014 *)

Crossrefs

Cf. A000081, A051491, A000055, A086308.

Sequence in context: A275473 A131805 A197694 * A103218 A319311 A107381

Adjacent sequences: A187767 A187768 A187769 * A187771 A187772 A187773

Keyword

nonn,cons

Author

Vaclav Kotesovec, Jan 04 2013

Status

approved