TITLE: Expression for $\sum\limits_{k=0}^n \frac1{k+1}\left\{ n \atop k\right\}$?
QUESTION [6 upvotes]: It is known that $\sum\limits_{k=0}^n \left\{ n \atop k\right\} k = \varpi(n+1) - \varpi(n)$. Any ideas for computing $\sum\limits_{k=0}^n \frac1{k+1}\left\{ n \atop k\right\}$ ? ($\left\{ n \atop k\right\}$ denotes the Stirling numbers of the second kind and $\varpi(n)$ the $n$-th Bell number)

REPLY [2 votes]: The error in leonbloy's approximation $\sum_{k=0}^n\frac1{k+1}\left\{ n \atop k\right\} =  B_{n-1} + E_n$ is exactly $$E_n = - \sum_{k=0}^{n-1}\left\{ n-1 \atop k\right\} \frac1{(k+1)(k+2)}.$$
Moreover, the asymptotic can be improved to $$\sum_{k=0}^n\frac1{k+1}\left\{ n \atop k\right\} \sim  B_{n-1} - B_{n-3},$$
(and probably further, for anyone who wants to continue the process below).

Theorem 4 of my paper "On Solutions to a General Combinatorial Recurrence" (Journal of Integer Sequences, 14 (9): Article 11.9.7, 2011), with $\left| {n \atop k} \right| = \left\{ n \atop k\right\}$ and $f(k,m) = \frac{1}{k+1}$ says that
$$\begin{align}\sum_{k=0}^n\frac1{k+1}\left\{ n \atop k\right\} &= \sum_{k=0}^{n-1}\left\{ n-1 \atop k\right\} (k+1) \frac1{k+1} - \sum_{k=0}^{n-1}\left\{ n-1 \atop k\right\} \frac1{(k+1)(k+2)} \\
&= B_{n-1} - \sum_{k=0}^{n-1}\left\{ n-1 \atop k\right\} \frac1{(k+1)(k+2)}. \end{align} $$
Applying Theorem 4 again, this time with $f(k,m) = \frac{1}{(k+1)(k+2)}$, yields (after some simplification)
$$\begin{align} &\sum_{k=0}^{n-1}\left\{ n-1 \atop k\right\} \frac1{(k+1)(k+2)} \\
&= \sum_{k=0}^{n-2}\left\{ n-2 \atop k\right\} \frac1{k+1} - \sum_{k=0}^{n-2}\left\{ n-2 \atop k\right\} \frac1{(k+1)(k+2)} - 2 \sum_{k=0}^{n-2}\left\{ n-2 \atop k\right\} \frac1{(k+1)(k+2)(k+3)}. \end{align}$$
The first term on the right-hand side dominates, and with leonbloy's approximation $$\sum_{k=0}^{n-2}\left\{ n-2 \atop k\right\} \frac1{k+1} \sim B_{n-3},$$
we get $$\sum_{k=0}^n\frac1{k+1}\left\{ n \atop k\right\} \sim  B_{n-1} - B_{n-3}.$$
By comparison with leonbloy's results (again, after taking logarithms)
n     exact      B_{n-1} - B_{n-3}     B_{n-1}
4    1.5114           1.3863            1.6094
8    6.7348           6.7154            6.7765
16  21.0308          21.0273           21.0475
32  57.5872          57.5866           57.5935