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A261875
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Decimal expansion of the coefficient 'gamma' (see formula) appearing in Otter's result concerning the asymptotics of T_n, the number of non-isomorphic rooted trees of order n.
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0
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2, 6, 8, 1, 1, 2, 8, 1, 4, 7, 2, 6, 7, 1, 1, 2, 2, 3, 8, 5, 7, 7, 3, 2, 8, 7, 8, 3, 7, 0, 3, 9, 3, 7, 0, 9, 3, 5, 4, 1, 7, 5, 3, 4, 7, 2, 0, 1, 1, 6, 1, 6, 6, 3, 5, 2, 7, 4, 9, 7, 0, 2, 5, 8, 8, 6, 4, 0, 2, 8, 4, 0, 3, 6, 5, 1, 6, 5, 3, 4, 5, 0, 6, 7, 2, 3, 9, 2, 0, 8, 5, 5, 8, 7, 7, 5, 9, 9, 1, 1
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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. 296.
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LINKS
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FORMULA
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Lim_{n->infinity} T_n*n^(3/2)/alpha^n = (beta/(2 Pi))^(1/3) = (1/(4 Pi alpha))^(1/2)*gamma, where alpha is A051491 and beta is A086308.
gamma = 2^(2/3)*Pi^(1/6)*beta^(1/3)*sqrt(alpha).
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EXAMPLE
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2.68112814726711223857732878370393709354175347201161663527497...
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MATHEMATICA
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digits = 100; max = 250; Clear[s, a]; 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}]; APrime[x_] := Sum[k*a[k]*x^(k-1), {k, 0, max}]; eq = Log[c] == 1 + Sum[A[c^-k]/k, {k, 2, max}]; alpha = c /. FindRoot[eq, {c, 3}, WorkingPrecision -> digits+5]; beta = (1+Sum[APrime[alpha^(-k)]/alpha^k, {k, 2, max}])^(3/2)/Sqrt[2*Pi]; gamma = 2^(2/3)*Pi^(1/6)*beta^(1/3) * Sqrt[alpha]; RealDigits[gamma, 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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