OFFSET
1,2
COMMENTS
E.g.f. A(x) = (1 - exp(-x)) o x/(1-x) o -log(1-x), a composition of functions such that the n-th iteration of A(x) equals (1 - exp(-x)) o x/(1-n*x) o -log(1-x).
EXAMPLE
E.g.f.: A(x) = x + 2*x^2/2! + 6*x^3/3! + 22*x^4/4! + 80*x^5/5! + 98*x^6/6! +...
Illustration of the initial iterations of A(x).
The 2nd iteration of A(x) equals:
A(A(x)) = 1 - (1-x)^(1/(1 + 2*log(1-x))), which begins:
A(A(x)) = x + 4*x^2/2! + 24*x^3/3! + 188*x^4/4! + 1780*x^5/5! +...
The 3rd iteration of A(x) equals:
A(A(A(x))) = 1 - (1-x)^(1/(1 + 3*log(1-x))), which begins:
A(A(A(x))) = x + 6*x^2/2! + 54*x^3/3! + 642*x^4/4! + 9420*x^5/5! +...
The 4th iteration of A(x) equals:
A(A(A(A(x)))) = 1 - (1-x)^(1/(1 + 4*log(1-x))), which begins:
A(A(A(A(x)))) = x + 8*x^2/2! + 96*x^3/3! + 1528*x^4/4! + 30200*x^5/5! +...
etc.
MATHEMATICA
With[{nn=30}, Rest[CoefficientList[Series[1-(1-x)^(1/(1+Log[1-x])), {x, 0, nn}], x]Range[0, nn]!]] (* Harvey P. Dale, Sep 22 2012 *)
PROG
(PARI) /* a(n, k) = the coefficient of x^n/n! in the k-th iteration: */
{a(n, k=1)=n!*polcoeff(1-(1-x+x*O(x^n))^(1/(1+k*log(1-x+x*O(x^n)))), n)}
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Dec 27 2011
STATUS
approved