A052138 Numerators of coefficients in function a(x) such that a(a(a(x))) = log (1+x).

1, -1, 1, -5, 103, -49, 2971, -34409, 10787, -567923, 4857119, -30312479, 7045653829, -77510407993, 262952596463, -58196505296117, 74362625639717, 172749391066639, -24672728226124829, -12265872397466432881, 1772477431815925044131

Offset

1,4

Comments

A(x)=sum(n>0 b(n)x^n/(3^(n-1)*n!), b(n)=T(n,1)*n!*3^(n-1) - integer.

References

W. C. Yang, Composition equations, preprint, 1999.

Links

Table of n, a(n) for n=1..21.

Formula

a(x)=sum_{n=1,2,3..} A052138(n)/A052139(n)*x^n. - R. J. Mathar, Jun 21 2007

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

Mathematica

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[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 *)

Prog

(Maxima)
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));
makelist(num(T(n, 1)), n, 1, 7); /* Vladimir Kruchinin, Mar 10 2012 */

Crossrefs

Cf. A052139. See also A048602, A048603, etc.

Sequence in context: A210900 A197799 A266903 * A308459 A142418 A377474

Adjacent sequences: A052135 A052136 A052137 * A052139 A052140 A052141

Keyword

sign,frac,easy,nice

Author

N. J. A. Sloane, Jan 22 2000

Extensions

More terms from R. J. Mathar, Jun 21 2007

Status

approved