Numerically evaluate a series involving Stirling numbers of the second kind How can I numerically evaluate the following for $n$ above one million:
$$ \sum_{k>=n} \sum_{i=0}^{n-1} (-1)^i {{n-1}\choose{i}} \left( \frac{n-1-i}{n} \right)^{k-1} k (k+1)$$?
I have tried changing the order of summation and the getting closed form for the geometric series in $k$ but the terms are large (even for $n = 100$) and I run into precision problems.

Too long for comments.
Starting from the point where @Phicar wrote
$$S_n=2\sum _{k= n}^\infty\left (\frac{n-1}{n}\right )^{k-1}\binom{k+1}{2}$$ we have
$$S_n=\sum_{k= n}^\infty k (k+1) \left(\frac{n-1}{n}\right)^{k-1}=(n-1)^{n-1} n^{3-n} (5 n-3)$$ and for large values of $n$
$$S_n=\frac 1e\left(5 n^3-\frac{n^2}{2}-\frac{n}{24}+\frac{1}{16}+\frac{95}{1152 n}+\frac{917}{11520
   n^2}+O\left(\frac{1}{n^3}\right)\right)$$ which seems to be quite different; however, none of these match the numbers that you provided as examples.
