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\title{Comment on `Automatic Central Limit Theorems ...'}
\author{Herb Wilf}
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\section{Better recurrence}
Recall that some years ago you were overly generous and credited
Nijenhuis and myself with inventing the algorithm for raising a
power series to the $n$th power, that appears in our book
\textit{Combinatorial Algorithms}. We pointed out that it was known
long before us, and Don Knuth traced it back to the Neanderthals.
Anyway that same algorithm gives a somewhat simpler recurrence for
your $f_r(n)$'s in this paper.

In fact, if
\[\left(\sum_{r\ge 0}\frac{F_r}{r!}z^r\right)^n=\sum_{r\ge
0}\frac{f_r(n)}{r!}z^r,\] then by logarithmic differentiation and
matching coefficients one finds that
\[f_r(n)=\sum_{0\le j<r}\left(n{r-1\choose j}-{r-1\choose j+1}\right)F_{j+1}f_{r-1-j}(n),\]
with $f_0=1$, $f_1=0$, which avoids the extra step of indefinite
summation that's in your present recurrence.




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