(The coefficients of *p*_{k}(*x*), incidentally, enumerate certain phylogenetic trees called Greg trees: [*x*^{j}] *p*_{k}(*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 *q*_{k}(*x*) = *p*_{k}(–*x*), we can prove by induction that for 0 ≤ *x* ≤ 1. Therefore *q*_{k}(*x*) decreases monotonically from *k*^{k – 1} to (*k* – 1)! as *x* goes from 0 to 1, for all *k*, *m* ≥ 1. It follows that

where the partial sums ...