Monotonic Exponential Generating Functions

SPECIES

$\displaystyle e^x = \sum_{n \geq 0} \dfrac{x^n}{n!} = 1 + x + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \dfrac{x^4}{4!} + \dfrac{x^5}{5!} + \cdots$

$\displaystyle \dfrac{1}{1-e^{-x}} = \sum_{n \geq 0} (-1)^n\,B_n^{-}\,\dfrac{x^{n-1}}{n!} \\ = \dfrac{1}{x} + \dfrac{1}{2} + \dfrac{1}{6}\,\dfrac{x}{2!} + \dfrac{1}{30}\,\dfrac{x^3}{4!} + \dfrac{1}{42}\,\dfrac{x^5}{6!} + \cdots$

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