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A052138 Numerators of coefficients in function a(x) such that a(a(a(x))) = log (1+x). 2
1, -1, 1, -5, 103, -49, 2971, -34409, 10787, -567923, 4857119, -30312479, 7045653829, -77510407993, 262952596463, -58196505296117, 74362625639717, 172749391066639, -24672728226124829, -12265872397466432881, 1772477431815925044131 (list; graph; refs; listen; history; text; internal format)
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 * A142418 A159523 A172116

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

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Last modified May 26 18:26 EDT 2017. Contains 287129 sequences.