A052139 Denominators of coefficients in function a(x) such that a(a(a(x))) = log (1+x).

1, 6, 18, 216, 9720, 9720, 1224720, 29393280, 18895680, 1984046400, 32736765600, 392841187200, 183849675609600, 4211828932147200, 25738954585344000, 8339421285651456000, 21265524278411212800, 437462213727316377600

Offset

1,2

References

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

Links

Table of n, a(n) for n=1..18.

Formula

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

Mathematica

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[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 *)

Crossrefs

Cf. A052138. See also A048602, A048603, etc.

Sequence in context: A214537 A052634 A059944 * A354019 A052682 A214592

Adjacent sequences: A052136 A052137 A052138 * A052140 A052141 A052142

Keyword

nonn,frac,easy,nice

Author

N. J. A. Sloane, Jan 22 2000

Extensions

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

Status

approved