Computing the n-th Bell number Bn

There are at least six ways of defining Bn:
1. $$B_n$$ is the number of partitions of a set with $$n$$ elements;
2. $$B_{n+1}=\displaystyle{\sum_{k=0}^n}\binom{n}{k}B_k$$, with $$B_0=1$$;
3. $$B_n=\displaystyle\sum_{k=1}^n\frac{k^n}{k!}\sum_{j=0}^{n-k}\frac{(-1)^j}{j!};$$
4. $$B_n=\displaystyle\frac{1}{e}\sum_{k=0}^\infty\frac{k^n}{k!};$$
5. Via the Aitken table $$A_{i,j}$$:

$$\displaystyle A_{1,1}=1,$$
$$\displaystyle A_{n,1}=A_{n-1,n-1}$$, $$\quad n\gt 1,$$
$$\displaystyle A_{n,k}=A_{n,k-1}+A_{n-1,k-1}, \quad 1\lt k\leq n$$.

Then $$B_n=A_{n,n},\quad 1\leq n$$;

$$A_{n,k}$$ is the number of partitions of $$\{1,\ldots,n+1\}$$ in which $$\{k+1\}$$ is the singleton with the largest entry in the partition. (See references (2) and (3).)

6. $$B_n=\displaystyle{\sum_{k=1}^n}S_{n,k}$$,
where $$S_{n,k}, 1\le n, 1\le k\le n$$ is the Stirling number of the second kind:
$$S_{n,1}=1, n\ge 1$$ and $$S_{n+1,k}=k\times S_{n,k} + S_{n,k-1}, \quad 2\leq k\leq n$$.
References:
1. OEIS: A000110
2. The largest singleton of set partitions, Yidong Sun, Xiaojuan Wu ( reference supplied by David Guichard)
3. Bell numbers (David Guichard)

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