TITLE: What's the lower bound of the sum $S(n) = \sum_{k=1}^n \prod_{j=1}^k(1-\frac j n)$?
QUESTION [16 upvotes]: If we have
$$
S(n) = \sum_{k=1}^n \prod_{j=1}^k(1-\frac j n)
$$
What the lower bound of $S(n)$ when $n\to\infty$?
PS: If I didn't make any mistake when I calculate $S(n)$, then it should be $\Omega(n)$.  But I don't know how to get it.

REPLY [11 votes]: This is $Q(n)-1$, where $Q(n)$ is a sum that appears repeatedly in Knuth's work and that I mention in my answer to your other question.  A slight adaptation of the  argument there shows that the dominant term is (in the notation of the answer) $T_n(0)$, and so we get 
$$Q(n) = S(n)+1 = \sqrt{\frac{\pi n}{2}} + O(1).$$
If you want a more precise asymptotic, you can check out Knuth's Art of Computer Programming, Vol. 1 (3rd ed.), Section 1.2.11.3 (again, as I mention in  my other answer).  He gives
$$Q(n) = S(n)+1 = \sqrt{\frac{\pi n}{2}} - \frac{1}{3} + \frac{1}{12} \sqrt{\frac{\pi }{2n}} - \frac{4}{135n} + O(n^{-3/2}).$$