Bound number of loopless multigraphs

Question

Let $d_1,\dots,d_n\geq 0$ be integers and let $G_{d_1,\dots,d_n}$ denote the set of loopless multigraphs with this degree sequence. I want to prove the following bound on the number of such multigraphs: $$ \# G_{d_1,\dots,d_n}\leq\prod_{i=1}^n \sqrt{d_i!}(n-1)^{\frac{d_i}{2}}. $$