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Converting a recursive definition to an explicit one

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 …