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

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

Author

Jean-François Alcover, Apr 19 2016

Keywords

• nonn
• cons

Examples

			1.47417268689787373633434182339755001284962360495558090802...
		

References

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

Links

• 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.

Crossrefs

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

Programs

Mathematica

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

Formula

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