Alternating Exponential Generating Functions

SECTIONS

$\displaystyle \dfrac{e^x}{(1\!+\!e^x)^{m+1}} \, \sum_{k=0}^{m-1} (-1)^k\,A_m(k)\,e^{kx} = \sum_{n \geq 0} \eta(-n\!-\!m)\,\dfrac{x^n}{n!}$

Eulerian Numbers

$\displaystyle \dfrac{t\!-\!1}{t-e^{(1-t)x}} = \sum_{n \geq 0} (-1)^n\,A_n(t)\,\dfrac{x^n}{n!} \\ = 1 - A_1(t)\,x + A_2(t)\,\dfrac{x^2}{2!} - A_3(t)\,\dfrac{x^3}{3!} + A_4(t)\,\dfrac{x^4}{4!} \pm \cdots$

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