I'm interested in formulas for $(a)_n$ with $n \in \mathbf{N}$: $$ (a)_n = \prod_{k=0}^{n-1}(a+k) $$ i.e. $$(a)_n = \frac{\Gamma(a+n)}{\Gamma(a)} $$ and the Bernoulli numbers $B_n$ given by the generating funcion: $$ \frac{x}{e^x-1} = \sum_{n=0}^\infty B_n \frac{x^n}{n!} $$ Since the relations between Riemann $\zeta$ and $\Gamma$ function I expect that there could be a formula that links 'easily' $B_n$ to $(a)_n$.
I was looking on classical calculus books and on the internet, there are tons of formulas but I can't find a something that links $B_n$ to $(a)_n$.
Thank you for the answers!