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The exponential generating function of Bernoulli number is $f(x)=x/(\exp(x)-1)$.

$x$ can be treated as species of 1-element set. Also $\exp(x)-1$ can be treated as nonempty set.

So, $f(x)*(\exp(x)-1)=x$ should have some combinatorial interpretation of exponential generating function.

Does anyone have any idea ?

ueir
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    The Bernoulli numbers necessarily don’t all have the same sign. It is sometimes possible to get combinatorial interpretations where signs are involved but it’s tricky because the cancellations involved aren’t necessarily obviously combinatorially meaningful. – Qiaochu Yuan Oct 28 '20 at 19:18

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