With $S(0)=0$ one has$$S(n)=-\sum_{k=1}^n\frac{F_k}{k!}$$where $F_k$ are the Fubini numbers (also known as ordered Bell numbers). The proof is contained in my comments above, given that the exponential generating function for these numbers is$$\sum_{n=0}^\infty\frac{F_n}{n!}x^n=\frac1{2-e^x}.$$There are many different expressions for $F_n$ at the links to OEIS and Wikipedia that I gave, they might be used to obtain some other alternative expressions for $S(n)$; it is difficult to say which ones are more explicit and which ones less.
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