OFFSET
0,2
COMMENTS
A recursion exists for coefficients, but is too complicated to use without a computer algebra system.
REFERENCES
W. C. Yang, Polynomials are essentially integer partitions, preprint, 1999
W. C. Yang, Composition equations, preprint, 1999
LINKS
W. C. Yang, Derivatives are essentially integer partitions, Discrete Math., 222 (2000), 235-245.
EXAMPLE
x - x^2/4 + x^3 * 5/48 + ...
MATHEMATICA
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[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 *)
CROSSREFS
KEYWORD
frac,nonn,nice
AUTHOR
Winston C. Yang (yang(AT)math.wisc.edu)
STATUS
approved