%I #22 Dec 16 2014 08:09:34
%S 1,4,48,96,3840,30720,13440,2064384,92897280,594542592,130799370240,
%T 1121137459200,40809403514880,816188070297600,48971284217856000,
%U 5484783832399872000,62160883433865216000,1918107260244983808000
%N Denominators of coefficients in function a(x) such that a(a(x)) = log(1+x).
%C A recursion exists for coefficients, but is too complicated to use without a computer algebra system.
%D W. C. Yang, Polynomials are essentially integer partitions, preprint, 1999
%D W. C. Yang, Composition equations, preprint, 1999
%H W. C. Yang, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00412-4">Derivatives are essentially integer partitions</a>, Discrete Math., 222 (2000), 235-245.
%e x - x^2/4 + x^3 * 5/48 + ...
%t n = 18; a[x_] = Sum[c[k] k! x^k, {k, 1, n}]; sa = Series[a[x], {x, 0, n}]; coes = CoefficientList[ ComposeSeries[sa, sa] - Series[Log[1+x],{x,0,n}], x] // Rest; eq = Reduce[((# == 0) & /@ coes)]; Table[c[k] k!, {k, 1, n}] /. First[Solve[eq, Table[c[k], {k, 1, n}]]] // Denominator (* _Jean-François Alcover_, Mar 28 2011 + upgrading by _Olivier Gérard_ *)
%t T[n_, m_] := T[n, m] = If[n == m, 1, (StirlingS1[n, m]*m!/n! - Sum[T[n, i]*T[i, m], {i, m+1, n-1}])/2]; a[n_] := T[n+1, 1] // Denominator; Table[a[n], {n, 0, 17}] (* _Jean-François Alcover_, Dec 16 2014, after _Vladimir Kruchinin_ *)
%Y Cf. A048607.
%K frac,nonn,nice
%O 0,2
%A Winston C. Yang (yang(AT)math.wisc.edu)