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%I #43 Sep 22 2023 16:06:19
%S 2,9,5,5,7,6,5,2,8,5,6,5,1,9,9,4,9,7,4,7,1,4,8,1,7,5,2,4,1,2,3,1,9,4,
%T 5,8,8,3,7,5,4,9,2,3,0,4,6,6,3,5,9,6,5,9,5,3,5,0,4,7,2,4,7,8,9,0,5,9,
%U 6,4,7,3,3,1,3,9,5,7,4,9,5,1,0,8,6,6,6,8,2,8,3,6,7,6,5,8,1,3,5,2,5,3
%N Decimal expansion of Otter's rooted tree constant: lim_{n->inf} A000081(n+1)/A000081(n).
%C A000055(n) ~ A086308 * A051491^n * n^(-5/2), A000081(n) ~ A187770 * A051491^n * n^(-3/2). - _Vaclav Kotesovec_, Jan 04 2013
%C Analytic Combinatorics (Flajolet and Sedgewick, 2009, p. 481) has a wrong value of this constant (2.9955765856). - _Vaclav Kotesovec_, Jan 04 2013
%D S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 295-316.
%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 S. R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/constant/otter/otter.html">Otter's Tree Enumeration Constants</a> [Broken link]
%H S. R. Finch, <a href="http://web.archive.org/web/20010207195939/http://www.mathsoft.com/asolve/constant/otter/otter.html">Otter's Tree Enumeration Constants</a> [Wayback Machine]
%H P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009, p. 481
%H Simon Plouffe, <a href="http://www.plouffe.fr/simon/constants/treegrowth.txt">Tree-growth constant to 1800 digits</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RootedTree.html">Rooted Tree</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Tree.html">Tree</a>
%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%e 2.95576528565199497471481752412319458837549230466359659535...
%t digits = 99; 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}]; eq = Log[c] == 1+Sum[A[c^-k]/k, {k, 2, max}]; alpha = c /. FindRoot[eq, {c, 3}, WorkingPrecision -> digits+5]; RealDigits[alpha, 10, digits] // First (* _Jean-François Alcover_, Sep 24 2014 *)
%Y Cf. A000081, A051492, A187770, A000055, A086308, A215978, A274082.
%K nonn,nice,cons
%O 1,1
%A _David Broadhurst_
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