%I #18 Aug 10 2021 11:09:42
%S 2,1,9,1,8,3,7,4,0,3,1,9,7,1,2,6,3,0,6,4,7,8,6,9,9,5,0,2,8,5,7,5,3,6,
%T 4,9,1,1,0,6,1,8,3,5,0,7,5,8,2,4,5,0,3,8,1,5,6,3,4,4,9,2,7,7,9,1,6,4,
%U 2,8,1,3,0,3,1,8,2,8,4,1,1,5,0,4,3,0,0,7,6,4,3,6,3,8,8,8,7,3,6,9
%N Decimal expansion of 'theta', the expected degree (valency) of the root of a random rooted tree with n vertices.
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.6 Otter's tree enumeration constants, p. 303.
%H A. Meir and J. W. Moon, <a href="http://dx.doi.org/10.4153/CJM-1978-085-0">On the altitude of nodes in random trees</a>, Canad. J. Math. 30(1978), 997-1015 Published:1978-10-01, page 1011.
%H E. M. Palmer and A. J. Schwenk, <a href="https://doi.org/10.1016/0095-8956(79)90073-X">On the Number of Trees in a Random Forest</a>, Journal of Combinatorial Theory, Series B, volume 27, number 2, October 1979, pages 109-121, see page 119 expected number of rooted trees in a rooted forest.
%F theta = 2 + Sum_{j>=1} T_j/(alpha^j*(alpha^j-1)), where T_j is A000081(j) and alpha A051491.
%e 2.19183740319712630647869950285753649110618350758245...
%t Clear[th]; digits = 100; m0 = 100; dm = 100; th[max_] := th[max] = (Clear[T, s, a]; T[0] = 0; T[1] = 1; T[n_] := T[n] = Sum[Sum[d*T[d], {d, Divisors[j]} ] * T[n-j], {j, 1, n-1}]/(n-1); 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]; 2+Sum[T[j]*1/(alpha^j*(alpha^j-1)), {j, 1, max}]); th[m0]; th[max = m0 + dm]; While[Print["max = ", max]; RealDigits[th[max], 10, digits] != RealDigits[th[max - dm], 10, digits], max = max + dm];theta = th[max]; RealDigits[theta, 10, digits] // First
%Y Cf. A000081 (T_n), A051491 (alpha), A272056 (variance).
%K cons,nonn
%O 1,1
%A _Jean-François Alcover_, Aug 09 2015