On Bell numbers of type $D$

In this paper, we will introduce Bell numbers $D(n)$ of type $D$ as an analogue to the classical Bell numbers related to all the partitions of the set $[n]$. Then based on a signed set partition of type $D$, we will construct the recurrence relations of Bell numbers $D(n)$. In addition, we deduce the exponential generating function for $D(n)$. Finally, we will provide an explicit formula for $D(n)$.