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I read about Bell numbers and I'm looking for a way to generate these numbers quickly for tests and exams. I know there is a recursively relation but it is not useful for big numbers .

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    See whether this helps you: http://fredrikj.net/blog/2015/08/computing-bell-numbers/ – 2017-02-22
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    @Rohan Shoot, I just used the information in that question. – 2017-02-22

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You can calculate them rapidly by using the bell triangle.

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    Okay , thank you but do you know any algebraically way ? – 2017-02-22
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    what do you mean "algebraically way"? – 2017-02-22
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    My mean is a formula – 2017-02-22
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    A closed formula for the bell numbers is not known. – 2017-02-22
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    Therefore there is no sequence like arithmetic or geometric so that represents bell numbers . Is it true ? But in the bell triangle we can calculate it easily . It's weird! – 2017-02-22
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    well, the bell triangle has coplexity $\mathcal O (n^2)$, but the calculations are easier for a human than the recursive solution without it. – 2017-02-22
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The exponential generating function of the Bell numbers is $e^{e^{x}-1}$, that is, the coefficient of $x^n/n! $ in the power series expansion of $e^{e^{x}-1}$ is the number of partitions of a set of $n $ elements.

You would like to see Section 1.6 in Generating Functionology by Herbert.S.Wilf for more information. Hope it helps.

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    but can you use this to generate them by hand? – 2017-02-22
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    Doesn't this imply that the $n$-th derivative of $e^{e^x-1}$ is $B_{n}$? at $x=0$. – 2017-02-22