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We know that the power series $f(x)= e^x -c \in \mathbb C[[x]]$ for $c \neq 1$, has a multiplicative inverse, since it's constant coefficient is non-zero. I was wondering whether the inverse is known in a closed form. Something like inverse of $\frac{e^x-1}{x}$, which is interesting as it generates the Bernoulli numbers.

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Wolfram Alpha can find the inverse in terms of $c$.

The denominators of each term are easy to express in closed form.

The polynomials in the numerators seem to follow Euler's triangle.

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