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%I #14 Dec 15 2015 10:44:50
%S 1,6,18,216,9720,9720,1224720,29393280,18895680,1984046400,
%T 32736765600,392841187200,183849675609600,4211828932147200,
%U 25738954585344000,8339421285651456000,21265524278411212800,437462213727316377600
%N Denominators of coefficients in function a(x) such that a(a(a(x))) = log (1+x).
%D W. C. Yang, Composition equations, preprint, 1999.
%F a(x)=sum_{n=1,2,3..} A052138(n)/A052139(n)*x^n. - _R. J. Mathar_, Jun 21 2007
%t max = 18; f[x_] := Sum[c[k]*x^k, {k, 0, max}]; c[0] = 0; c[1] = 1; s[1] = {}; coes = CoefficientList[ Series[ f[f[f[x]]] - Log[1 + x], {x, 0, max}], x]; eqns = Rest[ Thread[ coes == 0]]; Do[eqns = Rest[eqns] /. s[k]; s[k+1] = Solve[eqns[[1]], c[k + 1]][[1]], {k, 1, max-1}]; Denominator[ Table[c[k], {k, 1, max}] /. Flatten[ Table[s[k], {k, 1, max}]]] (* _Jean-François Alcover_, Oct 19 2011 *)
%t T[n_, m_] := T[n, m] = If[n == m, 1, 1/3*(StirlingS1[n, m]*m!/n! - Sum[T[k, m]*Sum[T[n, i]*T[i, k], {i, k, n}], {k, m+1, n-1}] - T[m, m]*Sum[T[n, i]* T[i, m], {i, m+1, n-1}])]; Table[T[n, 1] // Denominator, {n, 1, 21}] (* _Jean-François Alcover_, Dec 15 2015, after _Vladimir Kruchinin_ *)
%Y Cf. A052138. See also A048602, A048603, etc.
%K nonn,frac,easy,nice
%O 1,2
%A _N. J. A. Sloane_, Jan 22 2000
%E More terms from _R. J. Mathar_, Jun 21 2007
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