Alternating Exponential Generating Functions

SUPERSPECIES

$\displaystyle \dfrac{x}{e^x\!-\!1} = \sum_{n \geq 0} B_n^{-}\,\dfrac{x^n}{n!} \\ = 1 - \dfrac{1}{2}\,x + \dfrac{1}{6}\,\dfrac{x^2}{2!} - \dfrac{1}{30}\,\dfrac{x^4}{4!} + \dfrac{1}{42}\,\dfrac{x^6}{6!} - \dfrac{1}{30}\,\dfrac{x^8}{8!} \pm \cdots$

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