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 A261124 Decimal expansion of 'theta', the expected degree (valency) of the root of a random rooted tree with n vertices. 1
 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 (list; constant; graph; refs; listen; history; text; internal format)
 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. 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). Sequence in context: A206243 A316711 A187549 * A100945 A133399 A128751 Adjacent sequences:  A261121 A261122 A261123 * A261125 A261126 A261127 KEYWORD cons,nonn AUTHOR Jean-François Alcover, Aug 09 2015 STATUS approved

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Last modified March 29 15:16 EDT 2020. Contains 333107 sequences. (Running on oeis4.)