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A261124
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Decimal expansion of 'theta', the expected degree (valency) of the root of a random rooted tree with n vertices.
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2
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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
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OFFSET
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1,1
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REFERENCES
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Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.6 Otter's tree enumeration constants, p. 303.
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LINKS
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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.
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FORMULA
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theta = 2 + Sum_{j>=1} T_j/(alpha^j*(alpha^j-1)), where T_j is A000081(j) and alpha A051491.
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EXAMPLE
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2.19183740319712630647869950285753649110618350758245...
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MATHEMATICA
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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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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