Exact and simple expressions for the (inverse) of the $n$-th Bernoulli number

Of course, there are many known equations and approximations relating to the $$n$$-th Bernoulli number like $$B^{-{}}_m = \sum_{k=0}^m \sum_{v=0}^k (-1)^v \binom{k}{v} \frac{v^m}{k + 1}$$ or $$|B_{2 n}| \sim 4 \sqrt{\pi n} \left(\frac{n}{ \pi e} \right)^{2n}.$$ Yet despite the uppper equation is an exact formula for the $$n$$-th Bernoulli number, I was wondering whether

1. there is a simple product formula (or another easily understandable expression) for an infinite sequence of Bernoulli numbers with even index
2. there is a nice and easily understandable formula for the inverse for (an infinite sequence) of Bernoulli numbers with even index.
• On 2), I think you mean “a simple summation formula for the multiplicative inverse”. A formula for the functional inverse that only has to be accurate at the even Bernoulli numbers would be easier. – Matt F. Feb 6 at 9:23