Let $B_n$ denote the [Bernoulli numbers][1] and let $\phi=\frac{1+\sqrt{5}}2$ be the [golden ratio][2].

I encountered the following infinite sum and would like to ask:
>**Question.** Is this true? If so, any proof?
$$\sum_{k=1}^{\infty}\sum_{j=k}^{2k}\binom{k}{j-k}\frac{B_{j+1}}{j+1}
=\frac{2\,\log\phi}{1-2\phi}.$$

*Caveat.* Do not try reversing summations, it diverges! 

[1]: http://mathworld.wolfram.com/BernoulliNumber.html
[2]: https://en.wikipedia.org/wiki/Golden_ratio