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%I #33 Sep 22 2023 16:04:08
%S 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,
%T 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,
%U 5,4,0,0,4,2,0,4,7,6,4,6,0,5,3,7,9,8,6
%N Decimal expansion of Otter's asymptotic constant beta for the number of rooted trees.
%C A000081(n) ~ 0.439924012571 * alpha^n * n^(-3/2), alpha = 2.95576528565199497... (see A051491)
%D S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 5.6., p.296
%D D. E. Knuth, Fundamental Algorithms, 3d Ed. 1997, p. 396.
%H Vaclav Kotesovec, <a href="/A187770/b187770.txt">Table of n, a(n) for n = 0..1799</a>, (this constant was computed by David Broadhurst in November 1999)
%H Amirmohammad Farzaneh, Mihai-Alin Badiu, and Justin P. Coon, <a href="https://arxiv.org/abs/2309.09779">On Random Tree Structures, Their Entropy, and Compression</a>, arXiv:2309.09779 [cs.IT], 2023.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RootedTree.html">Rooted Tree</a>
%e 0.43992401257102530404090339143454476479808540794...
%t 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 *)
%Y Cf. A000081, A051491, A000055, A086308.
%K nonn,cons
%O 0,1
%A _Vaclav Kotesovec_, Jan 04 2013
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