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The Bernoulli numbers have many (as demonstrated here). Here is one property/characteristation:

$$\frac{t}{e^t-1}=\sum_{k=0}^\infty \frac{B_k}{k!} t^k$$

Conspicuously missing from the MathOverflow thread is any occurrence of Lambert's W function (or ProductLog). However, the wikipedia page has one tantalizing reference (see here).

Can anyone elaborate on this connection?

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