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Is there an explicit form for $a_x$ (whole numbers x) given that $a_x = \displaystyle\sum_{i=1}^{x-1} \binom{x-1}{i} a_i$?

I've listed out the first few terms:

for $x=0,1,2,3,4,5,6, 7$

we have $a_x =1, 1, 2, 5, 15, 52, 203, 877$ respectively which shows no obvious pattern, except growing extremely quickly.

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You are computing the Bell numbers.

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These are Bell numbers, A000110. The relatively explicit formula given in the Encyclopedia: $$a_n=\frac{2n!}{\pi e}\Im\left(\int_{0}^{\pi} e^{e^{e^{ix}}} \sin(nx) dx \right)$$

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