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%I #19 Dec 15 2015 10:45:01
%S 1,-1,1,-5,103,-49,2971,-34409,10787,-567923,4857119,-30312479,
%T 7045653829,-77510407993,262952596463,-58196505296117,74362625639717,
%U 172749391066639,-24672728226124829,-12265872397466432881,1772477431815925044131
%N Numerators of coefficients in function a(x) such that a(a(a(x))) = log (1+x).
%C A(x)=sum(n>0 b(n)x^n/(3^(n-1)*n!), b(n)=T(n,1)*n!*3^(n-1) - integer.
%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
%F a(n):=numerator(T(n,1)), T(n,m)=1/3*(stirling1(n,m)*m!/n!-sum(k=m+1..n-1, T(k,m)*sum(i=k..n, T(n,i)*T(i,k)))-T(m,m)*sum(i,m+1,n-1, T(n,i)*T(i,m))), n>m, T(n,n)=1. - _Vladimir Kruchinin_, Mar 10 2012
%t max = 21; 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}]; Numerator[ 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] // Numerator, {n, 1, 21}] (* _Jean-François Alcover_, Dec 15 2015, after _Vladimir Kruchinin_ *)
%o (Maxima)
%o T(n,m):=if n=m then 1 else 1/3*(stirling1(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));
%o makelist(num(T(n,1)),n,1,7); /* _Vladimir Kruchinin_, Mar 10 2012 */
%Y Cf. A052139. See also A048602, A048603, etc.
%K sign,frac,easy,nice
%O 1,4
%A _N. J. A. Sloane_, Jan 22 2000
%E More terms from _R. J. Mathar_, Jun 21 2007
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