Sum of Sequence of Factorials

Theorem
The sequence $S = \left\langle{s_n}\right\rangle$ defined as:
 * $\displaystyle s_n = \sum_{k \mathop = 1}^n k!$

begins:
 * $1, 3, 9, 33, 153, 873, 5913, 46 \, 233, 409 \, 113, 4 \, 037 \, 913, \ldots$