A261875 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.

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

Offset

1,1

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.6 Otter's tree enumeration constants, p. 296.

Links

Table of n, a(n) for n=1..100.

Formula

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).

Example

2.68112814726711223857732878370393709354175347201161663527497...

Mathematica

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

Crossrefs

Cf. A000055, A000081, A051491, A086308, A187770.

Sequence in context: A155003 A327279 A021377 * A084686 A040164 A086353

Adjacent sequences: A261872 A261873 A261874 * A261876 A261877 A261878

Keyword

cons,nonn

Author

Jean-François Alcover, Sep 04 2015

Status

approved