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

Author

Vaclav Kotesovec, Jan 04 2013

Keywords

• nonn
• cons

Comments

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

Examples

			0.43992401257102530404090339143454476479808540794...
		

References

• Steven 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)
• David Broadhurst, Resurgent Integer Sequences, Rutgers Experimental Math Seminar, Feb 06 2025; Slides.
• 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

Crossrefs

Cf. A000081, A051491, A000055, A086308.

Programs

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 *)