cp's OEIS Frontend

A272056 Decimal expansion of the variance of the degree (valency) of the root of a random rooted tree with n vertices.

%I A272056 #11 Aug 09 2021 07:23:39
%S A272056 1,4,7,4,1,7,2,6,8,6,8,9,7,8,7,3,7,3,6,3,3,4,3,4,1,8,2,3,3,9,7,5,5,0,
%T A272056 0,1,2,8,4,9,6,2,3,6,0,4,9,5,5,5,8,0,9,0,8,0,2,0,4,2,1,8,7,8,4,5,3,9,
%U A272056 1,3,7,3,9,6,6,5,0,0,9,3,8,7,0,2,8,1,3,6,7,2,8,6,6,6,4,0,2,7
%N A272056 Decimal expansion of the variance of the degree (valency) of the root of a random rooted tree with n vertices.
%D A272056 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.6 Otter's tree enumeration constants, p. 303.
%H A272056 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.
%F A272056 1 + Sum_{j>=1} T_j*(2alpha^j-1)/(alpha^j*(alpha^j-1)^2), where T_j is A000081(j) and alpha A051491.
%e A272056 1.47417268689787373633434182339755001284962360495558090802...
%t A272056 Clear[v]; digits = 98; m0 = 400; dm = 100; v[max_] := v[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]; 1 + Sum[T[j]*(2 alpha^j - 1)/ (alpha^j*(alpha^j - 1)^2), {j, 1, max}]); v[m0]; v[max = m0 + dm]; While[ Print["max = ", max]; RealDigits[v[max], 10, digits] != RealDigits[ v[max - dm], 10, digits], max = max + dm]; RealDigits[v[max], 10, digits] // First
%Y A272056 Cf. A000081 (T_n), A051491 (alpha), A261124 (expected degree).
%K A272056 nonn,cons
%O A272056 1,2
%A A272056 _Jean-François Alcover_, Apr 19 2016