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A261124
Decimal expansion of 'theta', the expected degree (valency) of the root of a random rooted tree with n vertices.
2
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, 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, 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
OFFSET
1,1
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.
E. M. Palmer and A. J. Schwenk, On the Number of Trees in a Random Forest, 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.
FORMULA
theta = 2 + Sum_{j>=1} T_j/(alpha^j*(alpha^j-1)), where T_j is A000081(j) and alpha A051491.
EXAMPLE
2.19183740319712630647869950285753649110618350758245...
MATHEMATICA
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
CROSSREFS
Cf. A000081 (T_n), A051491 (alpha), A272056 (variance).
Sequence in context: A371930 A316711 A187549 * A100945 A133399 A128751
KEYWORD
cons,nonn
AUTHOR
STATUS
approved