A lenghty limit, $$\lim_{ n \to \infty}-\sqrt{8}\cdot \frac{n!}{B_n}\sum_{j=0}^{n}\frac{(1-2^{1-j})(1-2^{1+j-n})B_{n-j}B_j}{4^j(n-j)!j!}=\pi$$
The limit seems to appraoches to $\pi$ but I am not able to shows it.
where $B_n$ is the Bernoulli number
Can anyone help to verify this limit is correct?