Original author of an exponential generating function for the Bernoulli numbers? The Bernoulli numbers were being used long before Bernoulli wrote about them, but according to Wikipedia, "The Swiss mathematician Jakob Bernoulli (1654–1705) was the first to realize the existence of a single sequence of constants B0, B1, B2, ... which provide a uniform formula for all sums of powers." Did he publish an exponential generating function as such for the series and was he the first to do so? If not, who published it first? According to Wikipedia again, Abraham de Moivre was the first to introduce the concept of generating functions per se in 1730.
This question is motivated by MSE-Q143499.
Let me try to make the question clearer so that responses won't involve the multitude of uses or properties of the Bernoulli numbers, which are fascinating, but not what I'm addressing by this question. 
Who first published
$$\displaystyle\frac{t}{e^t-1}=\sum B_n \frac{t^n}{n!}$$
as an encoding of the Bernoulli numbers?

I highly recommend the book Sources in the Development of Mathematics: Infinite Series and Products from the Fifteenth to the Twenty-first Century, by Ranjan Roy (Cambridge University Press, 2011).  Got it from the library a couple of weeks ago. It has almost $1000$ pages of treasures.
On page $23$, Roy writes: "In the early $1730$'s, Euler found a generating function for the Bernoulli numbers, apparently unaware that Bernoulli had already defined these numbers in a different way."  (The generating function is the one that you gave.)  Roy is very careful about sources, so it seems very likely that Euler was first.
The only explicit reference to a paper that I found is to De Seriebus Quibusdam Considerationes (Euler), apparently written in $1740$.
