I found this sum in an old math problems book and it asks me to find its closed form. And for the life of me I can't find. Here it is
$$\sum_{p=1}^{\infty}\frac{4^{p+1}\pi^{2p}}{(2p)!}|B_{2p}|\left(\frac{1}{2p+3}+\frac{1}{4p+2}-\frac{3}{4p+4}\right)$$
Any hits or suggestions are welcomed.