A261875 - OEIS

%I #7 Sep 05 2015 14:08:56

%S 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,

%T 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,

%U 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

%N 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.

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

%F 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.

%F gamma = 2^(2/3)*Pi^(1/6)*beta^(1/3)*sqrt(alpha).

%e 2.68112814726711223857732878370393709354175347201161663527497...

%t 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

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

%K cons,nonn

%O 1,1

%A _Jean-François Alcover_, Sep 04 2015