The number $f(n)=B(n)-\sum_{k=1}^n B(n-k) {n \choose k}$ counts certain partitions of $[n]$.Which partitions?
From Wiki, The Bell numbers satisfy a recurrence relation involving binomial coefficients: $B_{n+1}=\sum_{k=0}^n {n \choose k} B_k$. A different summation formula represents each Bell number as a sum of Stirling numbers of the second kind $B_n=\sum_{k=0}^n {n\brace k}$.
The exponential generating function of the Bell numbers is $B(x)=\sum_{n=0}^{\infty} \frac{B_n}{n!}x^n=e^{e^x-1}$. Stirling numbers can be calculated using ${n\brace m}=\frac{1}{k!}\sum_{i=0}^k(-1)^i{k \choose i}(k-i)^n$.
For this problem, am I supposed to plug in and calculate by the above operations? Thanks.
Hint:$$\sum_{k=0}^n \binom{n}{k} B_{n-k} = \sum_{k=0}^n \binom{n}{n-k} B_k = \sum_{k=0}^n \binom{n}{k} B_k = B_{n+1}$$