A261124 -id:A261124 - cp's OEIS frontend

A346916 Decimal expansion of the limit as N->oo of the mean number of singletons per forest in the rooted forests of N vertices.

5, 1, 1, 3, 0, 8, 7, 9, 9, 3, 4, 1, 2, 4, 0, 5, 9, 8, 9, 7, 1, 9, 9, 9, 9, 1, 8, 9, 6, 6, 7, 5, 7, 5, 9, 4, 8, 3, 6, 9, 5, 5, 8, 6, 7, 7, 5, 2, 4, 5, 9, 0, 9, 7, 2, 6, 8, 9, 1, 1, 5, 2, 9, 8, 2, 0, 1, 1, 3, 7, 3, 5, 4, 0, 6, 2, 3, 0, 8, 3, 1, 3, 9, 9, 1, 6, 9, 2, 4, 1, 7, 0, 8, 7, 7, 0, 9, 0, 9, 5, 7, 6, 8, 5, 1

Offset: 0

Kevin Ryde, Aug 07 2021

There are A000081(N+1) rooted forests of N vertices and A087803(N) singletons in those forests, so the present constant is S = lim_{N->oo} A087803(N) / A000081(N+1).

The respective asymptotic formulas for A000081 and A087803 show that S = 1/(d-1) where d=A051491 is the growth power of rooted trees and forests.

			0.5113087993412405989719999189667575...

Kevin Ryde, Table of n, a(n) for n = 0..1797

Cf. A051491 (rooted tree growth), A346915 (vpar mems).
Cf. A000081 (number of rooted forests), A087803 (total singletons).
Cf. A261124 (mean number of component trees).

Equals 1/(A051491 - 1).

Equals (A346915 - 2)/3.

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

Original entry on oeis.org

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, 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, 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

Offset: 1

Jean-François Alcover, Apr 19 2016

			1.47417268689787373633434182339755001284962360495558090802...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.6 Otter's tree enumeration constants, p. 303.

A. Meir and J. W. Moon, On the altitude of nodes in random trees, Canad. J. Math. 30(1978), 997-1015 Published:1978-10-01, page 1011.

Cf. A000081 (T_n), A051491 (alpha), A261124 (expected degree).

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

1 + Sum_{j>=1} T_j*(2alpha^j-1)/(alpha^j*(alpha^j-1)^2), where T_j is A000081(j) and alpha A051491.

cp's OEIS Frontend

A346916 Decimal expansion of the limit as N->oo of the mean number of singletons per forest in the rooted forests of N vertices.

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