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(The coefficients of pk(x), incidentally, enumerate certain phylogenetic trees called Greg trees: [xj] pk(x) is the number of oriented trees with j unlabeled nodes and k labeled nodes, where leaves must be labeled and unlabeled nodes must have at least two children. See J. Felsenstein, Systematic Zoology 27 (1978), 27–33; L. R. Foulds and R. W. Robinson, Lecture Notes in Math. 829 (1980), 110–126; C. Flight, Manuscripta 34 (1990), 122–128.) If qk(x) = pk(–x), we can prove by induction that for 0 ≤ x ≤ 1. Therefore qk(x) decreases monotonically from kk – 1 to (k – 1)! as x goes from 0 to 1, for all k, m ≥ 1. It follows that

where the partial sums ...

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