Does $\sum_{n=0}^{\infty}\frac{B_{n-1}}{n!}B_n(z)$ have closed form?

Question

I'm trying to evaluate the closed form of $$\sum_{n=0}^{\infty}\frac{B_{n-1}}{n!}B_n(z),$$ if there exists, when the generating function of the Bernoulli Polynomials is: $$\frac{t e^{xt}}{e^t-1}=\sum\limits_{n=0}^{\infty}B_n(x)\frac{t^n}{n!},$$ and $B_n=B_n(0)$. Thanks for your time.