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A272056
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Decimal expansion of the variance of the degree (valency) of the root of a random rooted tree with n vertices.
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1
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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
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OFFSET
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1,2
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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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FORMULA
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1 + Sum_{j>=1} T_j*(2alpha^j-1)/(alpha^j*(alpha^j-1)^2), where T_j is A000081(j) and alpha A051491.
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EXAMPLE
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1.47417268689787373633434182339755001284962360495558090802...
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MATHEMATICA
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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
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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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